This document contains a multi-part math problem involving trigonometric functions. It asks the reader to:
1) Calculate various trigonometric ratios for given angles.
2) Graph trigonometric functions like sin(x), cos(x), and tan(x) and identify their characteristics.
3) Derive equations to represent translated, scaled and shifted versions of trigonometric functions based on their graphs.
This document provides details about a meeting that took place on February 12, 2013. The meeting agenda included a discussion on project status and key activities over the past month. Action items were assigned to team members to complete tasks by the next meeting.
The document discusses Pascal's triangle and the binomial theorem. It explains how to use combinations to determine the coefficients when expanding binomial expressions. Various examples are worked through, such as writing the 7th row of Pascal's triangle and expanding (A + B)5. The document also derives a general formula for finding any term when expanding a binomial expression. Homework problems are assigned at the end.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
1. The document provides examples of evaluating composite functions using tables and graphs of component functions f(x) and g(x).
2. It gives the explicit equations and domains/ranges for several composite functions formed from basic polynomials and rational functions.
3. The examples show how to determine the composition f(g(x)) or g(f(x)) by applying each function in sequence based on their definitions.
This document provides details about a meeting that took place on February 12, 2013. The meeting agenda included a discussion on project status and key activities over the past month. Action items were assigned to team members to complete tasks by the next meeting.
The document discusses Pascal's triangle and the binomial theorem. It explains how to use combinations to determine the coefficients when expanding binomial expressions. Various examples are worked through, such as writing the 7th row of Pascal's triangle and expanding (A + B)5. The document also derives a general formula for finding any term when expanding a binomial expression. Homework problems are assigned at the end.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
1. The document provides examples of evaluating composite functions using tables and graphs of component functions f(x) and g(x).
2. It gives the explicit equations and domains/ranges for several composite functions formed from basic polynomials and rational functions.
3. The examples show how to determine the composition f(g(x)) or g(f(x)) by applying each function in sequence based on their definitions.
The document appears to be a collection of text messages between two individuals sent on December 10, 2012. The messages discuss plans to meet for lunch, then getting coffee or drinks afterwards. Logistics of locations and times are debated, as well as discussing other personal matters. A decision is finally made to meet for lunch at a specific restaurant at 1pm.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document is a log of a practice exam review on January 22, 2013. It contains time-stamped entries documenting the review from 2:11 PM to 3:18 PM, with notes on each page numbered 1 through 22.
El documento habla sobre la fotografía y la percepción de imágenes. Discusa que las fotografías van más allá de simplemente registrar hechos y pueden ser consideradas como creaciones intelectuales. También explica que las imágenes pueden clasificarse en categorías como documentales u objetivos periodísticos o sociales. Explora conceptos como la representación de algo a través de una imagen y cómo las imágenes pueden representar fenómenos del mundo real.
This document provides solutions to exercises about analyzing and graphing rational functions. Some key points summarized:
- The exercises involve identifying vertical and horizontal asymptotes, holes, and domains by factoring rational functions. Oblique asymptotes are also determined.
- Graphing technology is used to verify characteristics like asymptotes and holes, and determine zeros of related functions.
- Students are asked to match rational functions to their graphs based on identified characteristics like asymptotes, holes, and behavior near non-permissible values.
This document contains 15 multiple choice and free response questions about sinusoidal functions and their graphs. Key concepts covered include:
- Identifying amplitude, period, and phase shift from graphs of sinusoidal functions
- Writing equations to represent sinusoidal graphs in terms of sine and cosine
- Sketching transformed sinusoidal graphs (shifts, stretches, reflections)
- Finding amplitude, period, phase shift, and vertical/horizontal shifts from equations
- Relating sinusoidal equations to their real world applications like a roller coaster track
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
This document contains solutions to exercises from a pre-calculus lesson on trigonometric ratios for any angle in standard position. It provides the values of trigonometric ratios such as sine, cosine, tangent, cosecant, secant and cotangent for various angles in a table. It also shows sample calculations for determining coterminal angles and expressions to represent all coterminal angles for given angles.
This document contains solutions to checkpoint problems from a pre-calculus textbook chapter on trigonometry. It includes sketches of angles in standard position, determining coterminal angles, evaluating trigonometric ratios, solving problems involving terminal points of angles, converting between degrees and radians, and finding the area of a circular sector.
This document contains solutions to exercises involving radian measure. It includes sketches of angles in standard position, conversions between degrees and radians, calculations of trigonometric ratios, determining arc lengths and sector areas of circles based on central angles and radii, and calculations of angular velocity and linear distance traveled given rotational information.
This document discusses radian measure as an alternative to degrees for measuring angles. Some key points:
- 1 radian is defined as the angle subtended when the arc length along a circle is equal to the circle's radius.
- The circumference of a circle is equal to 2πr, which is equivalent to 2π radians or 360°.
- Common trigonometric ratios like sine, cosine, and tangent are given for radian measures of 30°, 45°, 60°, and 90°.
- Conversions between degrees and radians use the formulas: θ (degrees) = (θ radians) * 180/π and θ (radians) = (θ degrees) * π/180.
This document contains a practice test with 7 multiple choice and free response trigonometry problems. The problems cover topics like determining trigonometric ratios given angle measures, using trigonometric functions to model periodic data, and graphing trigonometric functions. The document provides step-by-step solutions and explanations for each problem.
This document discusses measuring angles and rotations. It defines key terms like angle, revolution, radian, and provides examples for converting between radians and degrees. Specifically, it explains that a full circle equals 2π radians or 360 degrees, and a half-circle equals π radians or 180 degrees. Several examples show how to calculate arc lengths for given angles in radians or degrees.
Trigonometry involves measuring angles and relationships between sides and angles of triangles. There are six trigonometric ratios - sine, cosine, tangent, cotangent, secant and cosecant - that relate the measures of sides and angles. Angles can be measured in degrees or radians and converted between the two units. Important trigonometric identities relate the ratios to each other and allow trigonometric functions of combined angles to be simplified.
This document discusses circular measures including radians, arc length of a circle, area of a sector of a circle, and finding the perimeter of a segment of a circle. Several examples and exercises are provided to illustrate how to:
1. Convert between degrees and radians.
2. Calculate arc lengths and angles given the radius and subtending angle.
3. Find the area of sectors given the radius and subtending angle.
4. Determine unknown values such as the radius or angle when given the sector area.
5. Calculate perimeters of circular segments.
Past SPM questions are also presented involving finding angles, radii, or arc lengths based on information provided about circles, arcs, or sectors.
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
Chapter 2 polygons ii [compatibility mode]Khusaini Majid
The document discusses properties of regular and irregular polygons, including:
- A regular polygon has equal side lengths and interior angles, while an irregular polygon does not.
- The interior angle plus exterior angle of any polygon equals 180 degrees.
- The sum of the exterior angles of any polygon is 360 degrees.
- The sum of the interior angles of an n-sided polygon is (n - 2) × 180 degrees.
- For a regular polygon, the interior angle is (n - 2) × 180/n degrees and the exterior angle is 360/n degrees.
The document defines various terms related to circle geometry such as radius, diameter, chord, tangent, arc, and angle. It then lists 10 rules of circle geometry pertaining to angles, chords, tangents, and polygons. Finally, it provides some practice questions and links for additional resources on circle geometry concepts.
The document appears to be a collection of text messages between two individuals sent on December 10, 2012. The messages discuss plans to meet for lunch, then getting coffee or drinks afterwards. Logistics of locations and times are debated, as well as discussing other personal matters. A decision is finally made to meet for lunch at a specific restaurant at 1pm.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document is a log of a practice exam review on January 22, 2013. It contains time-stamped entries documenting the review from 2:11 PM to 3:18 PM, with notes on each page numbered 1 through 22.
El documento habla sobre la fotografía y la percepción de imágenes. Discusa que las fotografías van más allá de simplemente registrar hechos y pueden ser consideradas como creaciones intelectuales. También explica que las imágenes pueden clasificarse en categorías como documentales u objetivos periodísticos o sociales. Explora conceptos como la representación de algo a través de una imagen y cómo las imágenes pueden representar fenómenos del mundo real.
This document provides solutions to exercises about analyzing and graphing rational functions. Some key points summarized:
- The exercises involve identifying vertical and horizontal asymptotes, holes, and domains by factoring rational functions. Oblique asymptotes are also determined.
- Graphing technology is used to verify characteristics like asymptotes and holes, and determine zeros of related functions.
- Students are asked to match rational functions to their graphs based on identified characteristics like asymptotes, holes, and behavior near non-permissible values.
This document contains 15 multiple choice and free response questions about sinusoidal functions and their graphs. Key concepts covered include:
- Identifying amplitude, period, and phase shift from graphs of sinusoidal functions
- Writing equations to represent sinusoidal graphs in terms of sine and cosine
- Sketching transformed sinusoidal graphs (shifts, stretches, reflections)
- Finding amplitude, period, phase shift, and vertical/horizontal shifts from equations
- Relating sinusoidal equations to their real world applications like a roller coaster track
This document contains sample solutions to checkpoint questions about combining functions. It provides the steps to:
1) Sketch the graph of y = f(x)/g(x) given the graphs of y = f(x) and y = g(x)
2) Write explicit equations for functions like g(x), h(x), and k(x) that satisfy an equation like f(x) = g(x) - h(x) - k(x)
3) Determine the domain and range of functions formed by combining basic functions using operations like addition, subtraction, multiplication, and division.
This document contains solutions to exercises from a pre-calculus lesson on trigonometric ratios for any angle in standard position. It provides the values of trigonometric ratios such as sine, cosine, tangent, cosecant, secant and cotangent for various angles in a table. It also shows sample calculations for determining coterminal angles and expressions to represent all coterminal angles for given angles.
This document contains solutions to checkpoint problems from a pre-calculus textbook chapter on trigonometry. It includes sketches of angles in standard position, determining coterminal angles, evaluating trigonometric ratios, solving problems involving terminal points of angles, converting between degrees and radians, and finding the area of a circular sector.
This document contains solutions to exercises involving radian measure. It includes sketches of angles in standard position, conversions between degrees and radians, calculations of trigonometric ratios, determining arc lengths and sector areas of circles based on central angles and radii, and calculations of angular velocity and linear distance traveled given rotational information.
This document discusses radian measure as an alternative to degrees for measuring angles. Some key points:
- 1 radian is defined as the angle subtended when the arc length along a circle is equal to the circle's radius.
- The circumference of a circle is equal to 2πr, which is equivalent to 2π radians or 360°.
- Common trigonometric ratios like sine, cosine, and tangent are given for radian measures of 30°, 45°, 60°, and 90°.
- Conversions between degrees and radians use the formulas: θ (degrees) = (θ radians) * 180/π and θ (radians) = (θ degrees) * π/180.
This document contains a practice test with 7 multiple choice and free response trigonometry problems. The problems cover topics like determining trigonometric ratios given angle measures, using trigonometric functions to model periodic data, and graphing trigonometric functions. The document provides step-by-step solutions and explanations for each problem.
This document discusses measuring angles and rotations. It defines key terms like angle, revolution, radian, and provides examples for converting between radians and degrees. Specifically, it explains that a full circle equals 2π radians or 360 degrees, and a half-circle equals π radians or 180 degrees. Several examples show how to calculate arc lengths for given angles in radians or degrees.
Trigonometry involves measuring angles and relationships between sides and angles of triangles. There are six trigonometric ratios - sine, cosine, tangent, cotangent, secant and cosecant - that relate the measures of sides and angles. Angles can be measured in degrees or radians and converted between the two units. Important trigonometric identities relate the ratios to each other and allow trigonometric functions of combined angles to be simplified.
This document discusses circular measures including radians, arc length of a circle, area of a sector of a circle, and finding the perimeter of a segment of a circle. Several examples and exercises are provided to illustrate how to:
1. Convert between degrees and radians.
2. Calculate arc lengths and angles given the radius and subtending angle.
3. Find the area of sectors given the radius and subtending angle.
4. Determine unknown values such as the radius or angle when given the sector area.
5. Calculate perimeters of circular segments.
Past SPM questions are also presented involving finding angles, radii, or arc lengths based on information provided about circles, arcs, or sectors.
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
Chapter 2 polygons ii [compatibility mode]Khusaini Majid
The document discusses properties of regular and irregular polygons, including:
- A regular polygon has equal side lengths and interior angles, while an irregular polygon does not.
- The interior angle plus exterior angle of any polygon equals 180 degrees.
- The sum of the exterior angles of any polygon is 360 degrees.
- The sum of the interior angles of an n-sided polygon is (n - 2) × 180 degrees.
- For a regular polygon, the interior angle is (n - 2) × 180/n degrees and the exterior angle is 360/n degrees.
The document defines various terms related to circle geometry such as radius, diameter, chord, tangent, arc, and angle. It then lists 10 rules of circle geometry pertaining to angles, chords, tangents, and polygons. Finally, it provides some practice questions and links for additional resources on circle geometry concepts.
This document appears to be a trigonometry review worksheet containing 24 problems involving trigonometric functions such as sine, cosine, tangent, cotangent and their inverses. The problems cover topics like using trigonometric functions to find angles and coordinates, converting between degrees and radians, solving trigonometric equations, expressing trigonometric functions in terms of other angles, and finding reference angles.
This document provides information about trigonometric equations including:
- The standard forms of angles in the four quadrants.
- Graphs of trigonometric functions like sine, cosine, and tangent.
- Considering trigonometric ratios like sine of 30° in different quadrants.
- Steps to solve trigonometric equations by isolating the trig ratio and determining the reference angle.
- Four types of trigonometric equations and examples of solving each type.
- Techniques for factorizing and using trigonometric identities to solve equations.
- Determining solutions within a given interval.
This document discusses radian and degree measure of angles. It covers terminology used to describe angles, converting between radian and degree measure, finding coterminal angles, and classifying angles by quadrant. Radian measure is defined as the measure of an angle whose terminal side intercepts an arc of length r on a circle of radius r. Common conversions between degrees and radians are also provided.
This module introduces the unit circle and trigonometric functions. It defines a unit circle as a circle with radius of 1 unit and discusses dividing the unit circle into congruent arcs. The module then covers converting between degrees and radians, defining angles intercepting arcs, and visualizing rotations along the unit circle. It concludes by discussing angles in standard position, quadrantal angles, and coterminal angles. Students are expected to learn key concepts like the unit circle, converting measures, and relating angles to arclengths and rotations.
Gyrocompass field calibration edited (download to see full features)Basyar Nor
This document summarizes calibration procedures for a gyrocompass and DGPS integrity checks. It describes taking gyrocompass readings and comparing them to known baselines to calculate calibration adjustments. It also outlines checking DGPS positioning by comparing coordinates for a known point to the DGPS readings. Transit methods are provided to circle a platform and calculate its center coordinates to check the DGPS system.
This document contains information about trigonometry including definitions of sine, cosine, and tangent ratios using right triangles. It discusses the four quadrants and corresponding reference angles. Examples are provided to determine the values of trigonometric functions for angles in different quadrants using the appropriate reference angle. Practice problems involve finding trig values, identifying quadrants, and determining reference angles.
The document summarizes key formulas and concepts for calculating angle measurements of polygons:
1) The formula for calculating the sum of the interior angles of any polygon with n sides is 180°(n-2).
2) The sum of the exterior angles of any polygon will always be 360° because if the vertices are pulled into the center it forms a circle.
3) For a regular polygon, the measurement of each interior angle can be calculated by taking the formula 180°(n-2) and dividing by n. The document provides examples of calculating interior, exterior, and total angle measurements using these formulas.
Plane and Applied Surveying 2
Traversing Theory Part
Traverse Computations
Definition
Types of Meridian
Applications of traversing
Bearings
Correction for observed angles (closed traverse)
Check angular Misclosure
Adjust angular Misclosure
Calculate adjusted bearings
Compute (E, N) for each traverse line
Coordinates.
-Traversing
Methods of conducting Traverse
1. Theodolite
2. Total Station
2. Compass
3. GPS
Bearings
Bearings
Bearing is the angle which a certain line make with a
certain meridian. Bearing with respect to true meridian is
called true bearings while magnetic bearing is the angle
which a line makes with respect to magnetic meridian.
There are two ways to represent the bearings,
Fore and back bearings
Whole circle bearing (W.C.B) ,(Azimuth)
Reduced Bearing (R.B) or quadrant bearing
6 The bearing of a line measured in the forward direction of survey line is called the ‘Fore Bearing’ (FB) of that line.
The bearing of the line measured in the direction opposite to the direction
of the progress of survey is called the ‘Back Bearing’ (BB) of the line.
BB= FB ± 180°
+ sign is applied when FB is < 180°
- sign is applied when FB is > 180°
1) Whole Circle Bearing (W.C.B) (Azimuth)
Is the bearing always measured from north in clockwise direction to a point.
Whole Circle Bearing (W.C.B) (Azimuth)
2) Reduced Bearing
Reduced bearing or Quadrant bearing is the angle which a line
makes from North or South Pole whichever may be near. The value of angle is from 0° to 90° , and are taken either clock wisely or anti clock wisely.
-Quadrant bearing
The difference between the whole circle bearing and quadrant
bearing are as follows.
-Example The following fore bearings were observed for lines, AB, BC, CD, and DE Determine their back bearings: • 145°, 285°, 65°, 215°
Example The Fore Bearing of the following lines are given Find the
Back Bearing.
(a) FB of AB= 310° 30’
(b) FB of BC= 145° 15’
(c) FB of CD = 210° 30’
(d) FB of DE = 60° 45’
Example:
Convert the following whole circle bearing to quadrant or
reduced bearings :
( i ) 42ᵒ 30’ ( ii ) 126ᵒ 15’
( iii ) 242ᵒ 45’ ( iv ) 328ᵒ10’
Example
Convert the following reduced bearings to whole circle
bearings:
( I ) N 65ᵒ 12’ E ( ii ) S 36ᵒ 48’ E
( iii ) S 38ᵒ 18’ W ( iv ) N 26ᵒ 32’ W
Closed Traverse
• Ends at a known point with known direction Geometrical Constraints
-Adjust the deflection angles
2-Interior angles Traverse
Interior angles are measured clockwise or counterclockwise between two adjacent lines on the inside of a closed polygon figure.
Example
The following traverse have five sides with five internal
angles. Find the angular misclosure and apply the angle
correction
-3-Exterior angle Traverse
Correction for observed angles (closed traverse)
Example:
IF ∑observed angles for traverse (ABCDA)= 360˚00′ 48″ find misclosure and correct the interior angles. Check Allowable Angle Misclosure
Prepared by:Asst. Prof. Salar K.Hussein
Erbil Polytechnic University
Geometric Transformation. A geometric rotation refers to the rotating of a figure around a center of rotation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
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