4.3 Equivalent Trigonometric Expressions

• These trigonometric identities will help you understand how equations are equal and you will apply your understanding of these trigonometric identities in questions asking for you to use equivalent trigonometric expressions. 
•  The identities mostly used in this section are Co function identities for angles in the 1st and 2nd Quadrant. 
• This means that you will use a specific trigonometric identity for the angle found in either the 1st or 2nd quadrant.

1st. Quadrant:
1. sin30º= cos60º, both approx. = 0.5000
2. sin50º= cos40º, both approx. = 0.7660

2nd. Quadrant:
1. -sin20º=cos110º, both approx. = -0.3420
2. sin100º=cos10º, both approx. = -0.1736
Key: degrees is substituted for the x value

We will come up with the following identities:
1st Quadrant:
1. sinx = cos((π/2)-x)
2. cosx = sin((π/2)-x)

2nd Quadrant:
1. sinx= cos(x+(π/2))
2. -cosx= sin(x+(π/2))

3rd Quadrant:

1. -sinx=cos(x+π)

2. -cosx=sin(x+π)

4rd Quadrant:

1. -sinx=cos(2π-x)

2. cosx=sin(2π-x)

 Pythagorean Identities

For angle θ at which the functions are defined:

(1)  cos²θ + sin²θ = 1
(2)  1 + tan²θ = sec²θ

(3)  1 + cot²θ = csc²θ

Dividing both sides of the first equation by cos²θ results in the second equation.  Similarly, dividing both sides of the first equation by sin²θ results in the third equation.

Reciprocal Identities

Cofunction Identities

Double-Angle Identities

(1)   sin 2θ = 2 sin θ cos θ

(2)   cos 2θ = 2 cos²θ  − 1
(3)   cos 2θ = 1 − 2 sin²θ

(4)   cos²θ = ½(1 + cos 2θ)
(5)   sin²θ = ½(1 − cos 2θ)

Sum & Difference Identities

• sin(α + β) = sin α cos β + sin β cos α
• sin(α − β) = sin α cos β − sin β cos α
• cos(α + β) = cos α cos β − sin α sin β
• cos(α − β) = cos α cos β + sin α sin β

Even-Odd Identities

• sin(−θ) = −sin(θ)
• cos(−θ) = cos(θ)
• tan(−θ) = −tan(θ)
• cot(−θ) = −cot(θ)
• sec(−θ) = sec(θ)
• csc(−θ) = −csc(θ)