Pythagorean Identities: Step-by-Step Guide
The Pythagorean identities connect the squares of trig functions: sin^2(x) + cos^2(x) = 1, 1 + tan^2(x) = sec^2(x), and 1 + cot^2(x) = csc^2(x). They come straight from the unit circle equation x^2 + y^2 = 1.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Pick the right identity. Match the trig functions in the problem to one of the three identities.\[\sin^2 x + \cos^2 x = 1\]
- Solve for the unknown. Put in the value you know and get the squared unknown by itself.\[\cos^2 x = 1 - \sin^2 x\]
- Take the square root. Remember that there is a plus and a minus answer. The quadrant tells you which to keep.\[\cos x = \pm \sqrt{1 - \sin^2 x}\]
- Choose the correct sign. Use the quadrant (or other information) given in the problem.\[\text{Quadrant}\Rightarrow\text{sign}\]
- Check. Put your values back into the identity.\[\sin^2 x + \cos^2 x \stackrel{?}{=} 1\]
Worked example
If sin(x) = 3/5 and x is in Quadrant II, find cos(x).
Step 1: Use sin^2(x) + cos^2(x) = 1.
Step 2: cos^2(x) = 1 - (3/5)^2 = 1 - 9/25 = 16/25.
Step 3: cos(x) = plus or minus 4/5.
Step 4: In Quadrant II, cosine is negative, so cos(x) = -4/5.
Step 5: Check: (3/5)^2 + (-4/5)^2 = 9/25 + 16/25 = 25/25 = 1. It works.
The answer is cos(x) = -4/5.
Tips and common mistakes
- Memorize all three identities. Each one is the best tool for its group of functions.
- The quadrant tells you the sign when you take a square root. Do not skip this step.
- 1 + tan^2 = sec^2 comes from dividing sin^2 + cos^2 = 1 by cos^2.
- 1 + cot^2 = csc^2 comes from dividing by sin^2.
- Use these identities to swap one trig function for another when you simplify.
- When solving equations, replace sin^2 with 1 - cos^2 to get an equation with just one trig function.
- Draw the reference triangle in the correct quadrant to see the signs.
- These identities are true for every x where the functions are defined.