The document defines trigonometric functions using the unit circle. It shows that the sine of an angle is equal to the y-coordinate of the point on the unit circle where the terminal side of the angle intersects, while the cosine is equal to the x-coordinate. The tangent is defined as the ratio of the sine to the cosine. Key properties discussed include the periodic nature of the trig functions with periods of 360 degrees or 2π radians, and whether functions are even or odd based on their behavior under negative inputs.
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
Trigonometric Function of General Angles LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Trigonometric Functions of Angles
Trigonometric Function Values
Could find the Six Trigonometric Functions
Learn the signs of functions in different Quadrants
Could easily determine the signs of each Trigonometric Functions
Solve problems involving Quadrantal Angles
Find Coterminal Angles
Learn to solve using reference angle
Solve problems involving Trigonometric Functions of Common Angles
Solve problems involving Trigonometric Functions of Uncommon Angles
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
Trigonometric Function of General Angles LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Trigonometric Functions of Angles
Trigonometric Function Values
Could find the Six Trigonometric Functions
Learn the signs of functions in different Quadrants
Could easily determine the signs of each Trigonometric Functions
Solve problems involving Quadrantal Angles
Find Coterminal Angles
Learn to solve using reference angle
Solve problems involving Trigonometric Functions of Common Angles
Solve problems involving Trigonometric Functions of Uncommon Angles
Are you looking for solution of Fourier Transform? Want to understand what is Complex number system ? Or trying to find what is the meaning of Euler's Equation ? Then this book is gold stone for you.
Thanks
Jaydeep shah (E.C)
Ahmedabad India
radhey04ec@gmail.com
trigonometric system lesson of math on how to. solve triangle the unit cirlce is the guide to find the exact value of a triangle,it is the foundation on how to rely the exact value of pi ..finding the sin the cosine the tangent the secant the cosecant and the cotangent
TIU CET Review Math Session 6 - part 2 of 2youngeinstein
College Entrance Test Review
Math Session 6 - part 2 of 2
FUNCTIONS
How to evaluate
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Composite functions
Trigonometric Functions
Pythagorean Theorem
30 60 90 triangle
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Exponential Functions
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This is a primer on some of the foundations of 3D math used in computer graphics programming. This is the version of the talk from CocoaConf Chicago 2015.
A rate of change is a rate that describes how one quantity changes in relation to another.
A constant rate of change is the rate of change of a linear relationship.
Essential Question How can you find the unit rate from a line on a graph that goes through the origin?
Objective understand slope as it relates to rate of change
Essential Question How can you find the unit rate from a line on a graph that goes through the origin?
Objective understand slope as it relates to rate of change
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A Strategic Approach: GenAI in EducationPeter Windle
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.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
3. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be So points on this circle must satisfy this equation. (1,0) (0,1) (0,-1) (-1,0)
4. Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? (1,0) (0,1) (0,-1) (-1,0) x = 1/2 You can see there are two y values. They can be found by putting 1/2 into the equation for x and solving for y . We'll look at a larger version of this and make a right triangle.
6. (1,0) (0,1) (0,-1) (-1,0) We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always 1 because it is a radius of the circle. Notice the sine is just the y value of the unit circle point and the cosine is just the x value.
7. (1,0) (0,1) (0,-1) (-1,0) We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions. So if I want a trig function for whose terminal side contains a point on the unit circle, the y value is the sine, the x value is the cosine and y / x is the tangent.
8. Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? 45 ° We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions. 45 ° 90 ° 0 ° 135 ° 180 ° 225 ° 270 ° 315° These are easy to memorize since they all have the same value with different signs depending on the quadrant.
9. Can you figure out what these angles would be in radians? The circle is 2 all the way around so half way is . The upper half is divided into 4 pieces so each piece is /4. 45 ° 90 ° 0 ° 135 ° 180 ° 225 ° 270 ° 315°
10. Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division? 30 ° We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x . 30 ° 90 ° 0 ° 120 ° 180 ° 210 ° 270 ° 330° You'll need to memorize these too but you can see the pattern. 60 ° 150 ° 240 ° 300 °
11. Can you figure out what the angles would be in radians? 30 ° It is still halfway around the circle and the upper half is divided into 6 pieces so each piece is /6. 30 ° 90 ° 0 ° 120 ° 180 ° 210 ° 270 ° 330° 60 ° 150 ° 240 ° 300 ° We'll see them all put together on the unit circle on the next screen.
12. You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly.
13. Let’s think about the function f ( ) = sin What is the domain? (remember domain means the “legal” things you can put in for ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function) . The range is: -1 sin 1 (1, 0) (0, 1) (-1, 0) (0, -1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?)
14. Let’s think about the function f ( ) = cos What is the domain? (remember domain means the “legal” things you can put in for ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function) . The range is: -1 cos 1 (1, 0) (0, 1) (-1, 0) (0, -1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?)
23. Look at the unit circle and determine sin 420 °. All the way around is 360 ° so we’ll need more than that. We see that it will be the same as sin 60° since they are coterminal angles. So sin 420 ° = sin 60°. In fact sin 780 ° = sin 60° since that is just another 360° beyond 420°. Because the sine values are equal for coterminal angles that are multiples of 360° added to an angle, we say that the sine is periodic with a period of 360° or 2 .
24.
25.
26. The cosine is also periodic with a period of 360° or 2 . We see that they repeat every so the tangent’s period is . Let's label the unit circle with values of the tangent. (Remember this is just y / x )
27. Reciprocal functions have the same period. PERIODIC PROPERTIES sin( + 2 ) = sin cosec( + 2 ) = cosec cos( + 2 ) = cos sec( + 2 ) = sec tan( + ) = tan cot( + ) = cot 1 (you can count around on unit circle or subtract the period twice.) This would have the same value as
28. Now let’s look at the unit circle to compare trig functions of positive vs. negative angles. Remember negative angle means to go clockwise
29. Recall from College Algebra that if we put a negative in the function and get the original back it is an even function .
30. Recall from College Algebra that if we put a negative in the function and get the negative of the function back it is an odd function .
31. If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also. EVEN-ODD PROPERTIES sin(- ) = - sin (odd) cosec(- ) = - cosec (odd) cos(- ) = cos (even) sec(- ) = sec (even) tan(- ) = - tan (odd) cot(- ) = - cot (odd)