Solving Trig Equations

Front

Define the sine function in terms of a right triangle.

Back

In a right triangle, the sine of an angle θ is defined as the ratio of the length of the opposite side to the length of the hypotenuse: \( \sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}} \.

Front

What is the range of the sine function?

Back

The range of the sine function is [-1, 1].

Front

Define the cosine function in terms of a right triangle.

Back

In a right triangle, the cosine of an angle θ is defined as the ratio of the length of the adjacent side to the length of the hypotenuse: \( \cos(θ) = \frac{\text{adjacent}}{\text{hypotenuse}} \.

Front

What is the range of the cosine function?

Back

The range of the cosine function is [-1, 1].

Front

What is the Pythagorean identity involving sine and cosine?

Back

The Pythagorean identity states that \( \sin^2(θ) + \cos^2(θ) = 1 \) for any angle θ.

Front

Solve for θ: \( \sin(θ) = -1 \) on the interval [0, 2π).

Back

θ = \( \frac{3π}{2} \).

Front

Solve for x: \( \cos(x) - \cos^2(x) = 0 \).

Back

x = 0, \( \frac{π}{2} \), \( \frac{3π}{2} \).

Front

Solve for x: \( \sin(x) \cos(x) + \sin(x) = 0 \).

Back

x = 0, π.

Front

Solve for θ: \( 2\cos^2(θ) - 1 = 0 \).

Back

θ = \( \frac{π}{4}, \frac{3π}{4}, \frac{5π}{4}, \frac{7π}{4} \).

Front

Solve for x: \( \sin^2(x) - 2\sin(x) - 3 = 0 \).

Back

x = \( \frac{3π}{2} \).