Derive an identity for cos(2theta), done fast in 56 seconds
To start, we write cos(2theta) as cos(theta + theta), which is simply an expression of the angle addition formula. Next, we use the identity that states cos(a)cos(b) - sin(a)sin(b) = cos(a+b) to expand the expression.
After multiplying cos(theta) with cos(theta) and sin(theta) with sin(theta), we get cos²(theta) - sin²(theta). From there, we substitute sin²(theta) with the Pythagorean identity 1 - cos²(theta), which gives us cos²(theta) - (1 - cos²(theta)).
Simplifying the expression, we get 2cos²(theta) - 1, which is the identity for cos(2theta)! We show all the steps in detail, so you can follow along easily.
🤔 Here are some common questions related to the topic
1️⃣ What is the identity for cos(2theta)?
Answer The identity for cos(2theta) is 2cos²(theta) - 1.
Example If theta equals pi/4, then cos(2theta) is equal to 2cos²(pi/4) - 1 = 1/2.
2️⃣ What is the angle addition formula?
Answer The angle addition formula states that cos(a+b) = cos(a)cos(b) - sin(a)sin(b).
Example If a = 30° and b = 60°, then cos(a+b) = cos(90°) = 0, and cos(a)cos(b) - sin(a)sin(b) = (sqrt(3)/2)(1/2) - (1/2)(1/2) = sqrt(3)/4 - 1/4 = (sqrt(3) - 1)/4.
3️⃣ What is the Pythagorean identity?
Answer The Pythagorean identity states that sin²(theta) + cos²(theta) = 1.
Example If theta = pi/6, then sin²(pi/6) + cos²(pi/6) = (1/4) + (3/4) = 1.
4️⃣ How do you derive the identity for sin(2theta)?
Answer To derive the identity for sin(2theta), use the angle addition formula for sin(a+b) and substitute sin²(theta) with the Pythagorean identity 1 - cos²(theta).
Example The identity for sin(2theta) is 2sin(theta)cos(theta).
5️⃣ What is the identity for tan(2theta)?
Answer The identity for tan(2theta) is (2tan(theta))/(1-tan²(theta)).
Example If theta equals pi/4, then tan(2theta) is equal to (2tan(pi/4))/(1-tan²(pi/4)) = 2/1 = 2.
6️⃣ How do you derive the identity for cos(3theta)?
Answer To derive the identity for cos(3theta), use the angle addition formula for cos(a+b+c) and simplify the expression.
Example The identity for cos(3theta) is 4cos³(theta) - 3cos(theta).
7️⃣ How do you derive the identity for sin(3theta)?
Answer To derive the identity for sin(3theta), use the angle addition formula for sin(a+b+c) and simplify the expression.
Example The identity for sin(3theta) is 3sin(theta) -
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