1. R I A N G L E

2. hypotenuse leg In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse a b c We’ll label them a, b, and c and the angles  and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.   First let’s look at the three basic functions. SINE COSINE TANGENT They are abbreviated using their first 3 letters opposite adjacent

3. We could ask for the trig functions of the angle  by using the definitions. a b c You MUST get them memorized. Here is a mnemonic to help you.   The old Indian word: SOHCAHTOA opposite adjacent SOHCAHTOA

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

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

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

7. The 30-60-90 Triangle In a 30-60-90 triangle the sides are in a ratio of 1- - 2 This means I can build a triangle with these lengths for sides 60° 30° 90° 2 1 We can then find the six trig functions of 30°or 60° using this triangle. Be sure to locate the angle you want before you find opposite or adjacent side opp 60° side opp 30° side opp 90° I used the triangle and did adjacent over hypotenuse of the 60° to get this but it is the cofunction of sine so this shows again that cofunctions of complementary angles are equal.

8. 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. You need to know all of the values in Table 3 on page 520 in your book and special triangles can help you with that. When directions say "Find the exact value", you must know these values not a decimal approximation that your calculator gives you.

9. 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)

10. It is important to note WHICH angle you are talking about when you find the value of the trig function. a b c  Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so Let's choose : 3 4 5 sin  = Use a mnemonic and figure out which sides of the triangle you need for sine. opposite hypotenuse tan  = opposite adjacent Use a mnemonic and figure out which sides of the triangle you need for tangent. 

11. You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.  This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle. 3 4 5  Oh, I'm acute! So am I!

12. 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 reciprocal of which and flip them.

13. a b c  As a way to help keep them straight I think, The "s" doesn't go with "s" and the "c" doesn't go with "c" so if we want secant, it won't be the one that starts with an "s" so it must be the reciprocal of cosine. (have to just remember that tangent & cotangent go together but this will help you with sine and cosine). 3 4 5 Let's try one: sec  = so cot  = Which trig function is this the reciprocal of?  h a so a o

14. SUMMARY OF METHODS FOR FINDING THE REMAINING 5 TRIG FUNCTIONS OF AN ACUTE ANGLE, GIVEN ONE TRIG FUNCTION. METHOD 1 1. Draw a right triangle labeling  and the two sides you know from the given trig function. 2. Find the length of the side you don't know by using the Pythagorean Theorem. 3. Use the definitions (remembered with a mnemonic) to find other basic trig functions. 4. Find reciprocal functions by "flipping" basic trig functions.