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LC Math 2 Adv – Deriving a Trigonometric Identity (LT 12)
An identity is an equation that is always true regardless of the value(s) of the variable(s). One such identity in
trigonometry is 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1, known as the Pythagorean Identity. We are going to prove this identity.
When we prove identities, we work only on the most complex side of the equation and transform it into equivalent
forms until we reach an equivalent expression that is exactly the same as the expression on the simpler side.
For example, 3(x - 4)-2x = x -12 is an identity. We can show this by transforming the more complex side,
3(x - 4)-2x , to show that one of its equivalent forms is x -12, as shown here:
3(𝑥 − 4) − 2𝑥 = 𝑥 − 12
3𝑥 − 12 − 2𝑥 =
3𝑥 − 2𝑥 − 12 =
𝑥 − 12 = 𝑥 − 12
We will be proving that 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1 is an identity. The left side, 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛2 𝜃, is clearly the more complex
side of the identity so we will be working to show that one of its equivalent forms is simply 1.
Let’s start with a diagram of a right triangle.
1. Label the side lengths of the triangle a, b, and c.
2. Label one of the acute angles of the triangle 𝜃.
3. What relationship must exist between a, b, and c?
4. Express 𝑠𝑖𝑛𝜃 and 𝑐𝑜𝑠𝜃 in terms of a, b, and c.
sinq =
; cosq =
5. Prove the identity. Take the information from #3 and #4 above as given.
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
Problem Solving
1. Determine which of the following statements are identities. If a statement is an identity, specify for which values
of x the equation is true. If a statement is not an identity, explain your reasoning.
a) 𝑠𝑖𝑛𝑥 = 𝑠𝑖𝑛(𝑥 + 360°)
b) 𝑡𝑎𝑛𝑥 = 1
c) 𝑠𝑖𝑛𝑥 = 𝑠𝑖𝑛(−𝑥)
d) 𝑐𝑜𝑠𝑥 = 𝑠𝑖𝑛(90 − 𝑥)
1
2. In a right triangle with acute angle of measure θ, 𝑠𝑖𝑛𝜃 = . Use the Pythagorean identity to determine the value
2
of 𝑐𝑜𝑠𝜃.
3. Confirm your solution to #2 using a different method.
7
4. In a right triangle with acute angle of measure θ, 𝑠𝑖𝑛𝜃 = 9. Use the Pythagorean identity to determine the value
of 𝑡𝑎𝑛𝜃.
𝑙
5. Let 𝑠𝑖𝑛𝜃 = 𝑚, where 𝑙, 𝑚 > 0. Express 𝑐𝑜𝑠𝜃 and 𝑡𝑎𝑛𝜃 in terms of 𝑙 and 𝑚.