This document summarizes trigonometry concepts including defining trig ratios (sine, cosine, tangent) for right triangles and their inverses. It explains how to use trig ratios to find missing sides or angles of right triangles by setting up equations. Examples are provided to calculate trig ratios in a triangle, find side lengths using trig ratios, and find a missing angle. Mnemonics like SOH-CAH-TOA are suggested to memorize the trig ratio relationships.
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
For any right triangle
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
The concept of trigonometric ratios is important in Maths trigonometry. Some of applications and procedures are discussed here for easy understanding the concept clarity.
Trigonometry is a field of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Though many modern students who pursue this career are unaware of its history, it has always been important to engineers. Pythagoras, the Pythagorean Theorem and Archimedes’ The Great Bridge are just a few of the timeless concepts that the mathematics of triangles can be applied to. Trigonometry has two primary components: arithmetic and geometry. Geometry describes geometric relationships between lines and angles. Arithmetic is the study of multiplication, division, integration, and multiplication.
This math presentation tutorial provides a basic introduction to trigonometry. It explains trigonometry ratios, how to evaluate it using the right triangle trigonometry and SOHCAHTOA. In trigonometry explains hypotenuse, adjacent and opposite side.
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
The concept of trigonometric ratios is important in Maths trigonometry. Some of applications and procedures are discussed here for easy understanding the concept clarity.
Trigonometry is a field of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Though many modern students who pursue this career are unaware of its history, it has always been important to engineers. Pythagoras, the Pythagorean Theorem and Archimedes’ The Great Bridge are just a few of the timeless concepts that the mathematics of triangles can be applied to. Trigonometry has two primary components: arithmetic and geometry. Geometry describes geometric relationships between lines and angles. Arithmetic is the study of multiplication, division, integration, and multiplication.
This math presentation tutorial provides a basic introduction to trigonometry. It explains trigonometry ratios, how to evaluate it using the right triangle trigonometry and SOHCAHTOA. In trigonometry explains hypotenuse, adjacent and opposite side.
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
7 use of pythagorean theorem cosine calculation for guide rightDePlaque
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Trigonometric Ratios Definition, Formulas Table and Problems.pdfChloe Cheney
Learn everything about trigonometric ratios, formulas, identities, tables and tips to memorize them quickly. Practice the given questions to grasp the concepts thoroughly.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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1. Trigonometry
The student is able to (I can):
For any right triangle
• Define the sine, cosine, and tangent ratios and their
inverses
• Find the measure of a side given a side and an angle
• Find the measure of an angle given two sides
• Use trig ratios to solve problems
2. By the Angle-Angle Similarity Theorem, a right triangle with a
given acute angle is similar to every other right triangle with
the same acute angle measure. This means that the ratios
between the sides of those triangles are always the same.
Because these ratios are so useful, they were given names:
sinesinesinesine, cosinecosinecosinecosine, and tangenttangenttangenttangent. These ratios are used in the study
of trigonometrytrigonometrytrigonometrytrigonometry.
3. sine of ∠A
cosine of ∠A
tangent of ∠A
AAAA
hypotenuse
adjacent
opposite
∠
= =
leg opposite
sin
hypotenuse
A
A
leg adjacent to
cos
hypotenuse
A
A
∠
= =
leg opposite
tan
leg adjacent to
A
A
A
∠
= =
∠
4. We can use the trig ratios to find either missing sides or
missing angles of right triangles. To do this, we will set up
equations and solve for the missing part. In order to figure
out the sine, cosine, and tangent ratios, we can use either a
calculator or a trig table.
5. To use the Nspire calculator to find tan 51°:
• From a New Document, press the µ key:
• Use the right arrow key (¢) to select tan and press ·:
6. • Type 5I and hit ·:
To use the calculator on your phone:
• Turn your phone landscape to access the scientific
calculator.
• Depending on your phone, you will either teither teither teither type the angle in
first and select tan, orororor select tan and then type in the
angle.
7. To find an angle, we use the inverseinverseinverseinverse trig functions (you will
sometimes hear them referred to as arcsine, arccosine, and
arctangent). On your calculator, these are listed as sin–1,
cos–1, and tan–1.
Ex. Find :
Press the µ button, and then the ¤ arrow to select sin–1.
Then enter 8p17·. You should get 28.07…
This means that the angle opposite a leg of 8 with a
hypotenuse of 17 will measure around 28˚.
1 8
sin
17
−
8. You will be expected to memorize these ratio relationships.
There are many hints out there to help you keep them
straight. The most common is SOHSOHSOHSOH----CAHCAHCAHCAH----TOATOATOATOA , where
A mnemonic I like is “Some Old Hippie Caught Another
Hippie Trippin’ On Acid.”
Or “Silly Old Hitler Couldn’t Advance His Troops Over Africa.”
pp
in
yp
O
S
H
=
dj
os
yp
A
C
H
=
pp
an
dj
O
T
A
=
9. Examples
I. Use the triangle to find the following ratios.
1. sin B = _____
2. cos B = _____
3. tan B = _____
A
B
C
8
15
17
10. Examples
I. Use the triangle to find the following ratios.
1. sin B = _____
2. cos B = _____
3. tan B = _____
A
B
C
8
15
17
8
17
15
17
8
15
11. II. Find the lengths of the sides to the nearest tenth.
1.
2.
x (opp)
15
(adj)
58°
26
(hyp)
x
(adj)
46°
tan58
15
15tan58
24.0
° =
= °
≈
x
x
cos46
26
26cos46
18.1
° =
= °
≈
x
x