Sect 4 4 Trig Functions At Any Angle

Sect. 4. 4 Trig Functions: At Any Angle

Trig Functions of Any Angle In 4. 3, we looked at the definitions of the trig functions of acute angles of a right triangle. In this section, we will expand upon those definitions to include ANY angle. We will be studying angles that are greater than 90° and less than 0°, so we will need to consider the signs of the trig functions in each of the quadrants.

Definitions of Trig Functions of Any Angle Let be an angle in standard position with (x, y) a point on the terminal side of and or y (x, y) r x

Ex 1 in textbook: Let (– 3, 4) be a point on the terminal side of. Find the value of the six trig functions of. Solution: x = -3, y = 4, still need r

Let’s do one together Let (-12, -5) be a point on the terminal side of . Find the exact values of the six trig functions of . First we must find the value of r : y x -12 r 13 (-12, -5) -5

You Try… Let (-3, 7) be a point on the terminal side of . Find the value of the six trig functions of .

Evaluating Trig Functions When Given a Point To find the value of a trig function when given a point… 1. Calculate r using x 2 + y 2 = r 2. Use the definition of the trig functions to write the ratios using the appropriate x, y and r values. 3. Depending upon the quadrant in which lies, use the appropriate sign (+ or –).

The Signs of the Trig Functions Since the radius is always positive (r > 0), the signs of the trig functions are dependent upon the signs of x and y. Therefore, we can determine the sign of the functions by knowing the quadrant in which the terminal side of the angle lies.

Signs of Trig Functions

A trick to remember where each trig function is POSITIVE: All Students Take Calculus Translation: S T A C A = All 3 functions are positive in Quad 1 S= Sine function is positive in Quad 2 T= Tangent function is positive in Quad 3 C= Cosine function is positive in Quad 4 **Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, …

Ex 2 in textbook: Given and , find . Solution: Note lies in Quad III because that is the only quad in which sine is – and tangent is +. By letting , you can let y = -2 and r = 3.

Let’s do one together Given and , find the values of the five other trig function of . Solution First, determine the quadrant in which lies. Since the cosine is negative and the cotangent is positive, we know that lies in Quadrant _____.

Example (cont) Now we can find the values of the remaining trig functions:

You Try… Given and five other trig functions of . , find the values of the

Reference Angles The values of the trig functions for non-acute angles (Quads II, IV) can be found using the values of the corresponding reference angles. Let be an angle in standard position. Its reference angle is the acute angle ‘ formed by the terminal side of and the horizontal axis.

Example (like Ex 4 in TB) Find the reference angle for Solution . y By sketching in standard position, we see that it is a 3 rd quadrant angle. To find , you would subtract 180° from 225 °. x

Let’s do one together… Find the reference angle for . Solution To find , you would add -325° to 360°. You Try… Find the reference angles for the following angles. 1. 2. 3.

So what’s so great about reference angles? Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies. For example, find the exact value of In Quad 3, sine is negative . 45° is the ref angle

Let’s do one together… Give the exact value of the trig function (without using a calculator). 1. 2.

You Try… Evaluate each trigonometric function. 1. 2.

Evaluate Trig Functions Using Reference Angles To find the value of a trig function for any common angle 1. Determine the quadrant in which the angle lies. 2. Determine the reference angle by subtracting from the value of the horizontal axis that it is closest to. 3. Use one of the special triangles to determine the function value for the reference angle. 4. Depending upon the quadrant in which lies, use the appropriate sign (+ or –).

Closure Explain how to find the 6 trig functions of any angle. You may use the following as an example. y x 22