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Difficulty: EasyTrigonometric Ratios and Identities

An acute angle θ\theta satisfies the equation sin⁡(θ)=0.6\sin(\theta) = 0.6. What is the value of cos⁡(θ)\cos(\theta)?

Answer: 0.8

Answer

The value of cos⁡(θ)\cos(\theta) is 0.8.
Using the fundamental Pythagorean trigonometric identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1, we substitute the given value sin⁡(θ)=0.6\sin(\theta) = 0.6 to obtain (0.6)2+cos⁡2(θ)=1(0.6)^2 + \cos^2(\theta) = 1, which simplifies to 0.36+cos⁡2(θ)=10.36 + \cos^2(\theta) = 1. Subtracting 0.360.36 from both sides yields cos⁡2(θ)=0.64\cos^2(\theta) = 0.64. Taking the positive square root because θ\theta is an acute angle gives cos⁡(θ)=0.8\cos(\theta) = 0.8.

Step-by-Step Solution

1
State the Pythagorean trigonometric identity
sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1
This identity relates the sine and cosine of any angle.
2
Substitute the value of sin⁡(θ)\sin(\theta) into the identity
(0.6)2+cos⁡2(θ)=1(0.6)^2 + \cos^2(\theta) = 1
We are given that sin⁡(θ)=0.6\sin(\theta) = 0.6.
3
Solve for cos⁡2(θ)\cos^2(\theta)
cos⁡2(θ)=0.64\cos^2(\theta) = 0.64
Subtracting 0.360.36 from 11 isolates the squared cosine term.
4
Take the square root of both sides
cos⁡(θ)=0.8\cos(\theta) = 0.8
Since θ\theta is an acute angle, the value of cos⁡(θ)\cos(\theta) must be positive.

Key Concept

Pythagorean Identity
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Topics

Trigonometric Ratios and Identities

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