4.4 Trigonometric Functions of any Angle

Definitions of Trigonometric Functions of Any Angle:

Let < θ be any angle in standard position with (x, y) a point on the terminal side of < θ

^{2}= x

^{2}+ y

^{2}

so therefore:

r = the square root of (x^{2} + y^{2}) - and cannot equal zero.

sin θ = *y*/*r*

csc θ = *r*/*y* , *y* cannot equal 0.

cos θ = *x*/*r*

sec θ = *r*/*x* , *x* cannot equal 0.

tan θ = *y*/*x* , *x* cannot equal 0.

cot θ = *x*/*y* , *y* cannot equal 0.

**Example**: the given point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle.

1. (5, -12) When you plot this it is in the IV quadrant. Using Pythagorean theorem,

(5)^{2} + (-12)^{2} = r^{2}

25 + 144 = r^{2}

169 = r^{2}

13 = r

so you see the hypotenuse is 13. Therefore the values of the six trigonometric functions are:

sin θ = -12/13

csc θ = 13/-12

cos θ = 5/13

sec θ = 13/5

tan θ = -12/5

cot θ = -5/12

Reference Angles:

Let θ be an angle in standard position. Its reference angle is the acute angle θ’ formed by the terminal side of θ and the horizontal axis.

**Quadrant II **

θ’ = π - θ (radians)

θ’ = 180° - θ (degrees)

**Quadrant III **

θ’ = θ - π (radians)

θ’ = θ - 180° (degrees)

**Quadrant IV**

θ’ = 2π - θ (radians)

θ’ = 360° - θ (degrees) **Example**: Find the reference angle:

1. θ = 57° θ’ = 57°

2. θ = 125°

θ’ = 180° - 125° = 55°

3. θ = 295°

θ’ = 360° - 295° = 65°

4. θ = 2π/3

θ’ = π - 2π/3 = π/3

5. θ = 5π/6

θ’ = π- 5π/6 = π/6

6. θ = 11π/6

θ’ = 2π - 11π/6 = π/6

Evaluating Trigonometric Functions of any angle:

To find the value of a trigonometric function of any angle θ:

1. Determine the function value for the associated reference angle θ’.

2. Depending on the quadrant in which θ lies, prefix the appropriate sign to the function value.

By using reference angles and the special angles discussed in the previous section, you can greatly extend the scope of exact trigonometric values.

**Example**:

Evaluate the trigonometric function exactly:

1. sin -20π/3 = sin -2π /3 = sin 4π/3 =(since 4π/3 is in the III quadrant) = -sin(ref <>

2. cos( -300°) = cos 300 = (ref <) = cos 60° = ½

3. tan (- 495°) = tan (-135) = (ref <) = tan 45° = 1