The Trigonometric Functions

1. The Trigonometric Functions

2. First let’s look at the three basic trigonometric functions SINE (x, y) r COSINE TANGENT  hypotenuse They are abbreviated using their first 3 letters Let’s look at an angle  in standard position whose terminal side contains the point (x, y). Let r be the distance from the origin to the point (x, y). r can be found using the distance formula. The three basic trigonometric functions are defined as follows:

3. There are three more trig functions. They are called the reciprocal functions because they are reciprocals of the first three functions. Oh yeah, this means to flip the fraction over. Like the first three trig functions, these are referred to by the first three letters except for cosecant since it's first three letters are the same as for cosine. Best way to remember these is learn which is the reciprocal of which and flip them over.

4. RECIPROCAL IDENTITIES Based on the fact that these 3 trig functions are reciprocals of the three basic ones, they are called the reciprocal identities.

5. Find the values of the six trigonometric functions of the angle  in standard position whose terminal side passes through the point (4, -5) Often the preferred way to leave the answer is with a rationalized denominator  r (4, -5)

6. An angle whose terminal side is on an axis is called a quadrantal angle. (0, 1) A 90° angle is a quadrantal angle. To find the trig functions of 90°, choose a point on the terminal side. 1 90°

7. To fill in the following table of quadrantal angles use the graph below. Start with 0° going down. Figure out the answer and then click the mouse to see if you are right. (0, 1) (1, 0) (-1, 0) (0, -1) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

8. All trig functions positive In quadrant I both the x and y values are positive so all trig functions will be positive  Let's look at the signs of sine, cosine and tangent in the other quadrants. Reciprocal functions will have the same sign as the original since "flipping" a fraction over doesn't change its sign. sin is +cos is -tan is - In quadrant II x is negative and y is positive.  We can see from this that any trig function that requires the x value will then have a negative sign on it.

9. In quadrant III, x is negative and y is negative. The r is always positive so if we have either x or y with hypotenuse we'll get a negative. If we have both x and y the negatives will cancel  sin is -cos is -tan is + In quadrant IV, x is positive and y is negative .  So any functions using y will be negative. sin is -cos is +tan is -

10. To help remember these signs we look at what trig functions are positive in each quadrant. sin is +cos is -tan is - All trig functions positive S A T C sin is -cos is -tan is + sin is -cos is +tan is - Here is a mnemonic to help you remember. (start in Quad I and go counterclockwise) Students All Take Calculus

11. Computing the Values of Trig Functions of Acute Angles USING SPECIAL TRIANGLES

12. In a 45-45-90 triangle the sides are in a ratio of 1- 1- The 45-45-90 Triangle This means I can build a triangle with these lengths for sides (or any multiple of these lengths) (1, 1) We can then find the six trig functions of 45° using this triangle. 45° rationalized 1 45° 90° 1 You can "flip" these to get other 3 trig functions

13. You are expected to know exact values for trig functions of 45°. You can get them by drawing the triangle and using sides. What is the radian equivalent of 45°? You also know all the trig functions for /4 then. 45° 1 45° 90° reciprocal of cos 1

14. The 30-60-90 Triangle side opp 60° In a 30-60-90 triangle the sides are in a ratio of 1- - 2 side opp 90° side opp 30° This means I can build a triangle with these lengths for sides We can then find trig functions of 60° using this triangle. 30° 2 60° 90° 1

15. The 30-60-90 Triangle side opp 60° In a 30-60-90 triangle the sides are in a ratio of 1- - 2 side opp 90° side opp 30° We can draw the triangle so the 30° angle is at the bottom. We can then find trig functions of 30° using this triangle. 60° 2 1 90° 30°

16. What this means is that if you memorize the special triangles, then you can find all of the trig functions of 45°, 30°, and 60° which are common ones you need to know. You also can find the radian equivalents of these angles. When directions say "Find the exact value", you must know these values not a decimal approximation that your calculator gives you.

17. Here is a table of sines and cosines for common angles. You can get these by drawing the special triangles, but notice the pattern.

18. Using a Calculator to Find Values of Trig Functions If we wanted sin 38° we could not use the previous methods to find it because we don't know the lengths of sides of a triangle with a 38° angle. We will then use our calculator to approximate the value. You can simply use the sin button on the calculator followed by (38) to find the sin 38° A word to the wise: Always make sure your calculator is in the right mode for the type of angle you have (degrees or radians). If there is not a degree symbol then you know the angle is in radians.

19. Using a Calculator to Find Values of Reciprocal Trig Functions If we wanted csc (/5) we use our calculator to approximate the value remembering that cosecant is the reciprocal function of sine so is 1 over sine. You can simply put in 1 divided by sin followed by (/5) to find the csc /5 Make sure you are in radian mode and that you put the /5 in parenthesis.