Solving Trigonometric Equations II

Front

Define the sine function in a right triangle.

Back

The sine function (sin) is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

Front

What is the cosine function in terms of a right triangle?

Back

The cosine function (cos) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Front

What is the tangent function?

Back

The tangent function (tan) is defined as the ratio of the sine to the cosine, or the ratio of the opposite side to the adjacent side.

Front

What is the Pythagorean identity?

Back

The Pythagorean identity states that sin²(x) + cos²(x) = 1 for any angle x.

Front

How do you solve the equation 2sin x cos x = √2 cos x?

Back

Factor out cos x: cos x(2sin x - √2) = 0. Set each factor to zero: cos x = 0 or 2sin x - √2 = 0.

Front

What is the general solution for sin x = k?

Back

The general solution is x = arcsin(k) + 2nπ or x = π - arcsin(k) + 2nπ, where n is any integer.

Front

What does it mean to solve trigonometric equations on the domain [0, 2π)?

Back

It means to find all solutions within the interval from 0 to 2π, not including 2π.

Front

How do you solve the equation cos x + 2 = 3 cos x?

Back

Rearrange to get 2 = 2 cos x, leading to cos x = 1, which gives x = 0.

Front

What is the solution set for 4sin²x = 3?

Back

Solve for sin²x = 3/4, then sin x = ±√(3/4). The solutions are x = π/3, 2π/3, 4π/3, 5π/3.

Front

How do you solve 1/2sec x - 1 = 0?

Back

Rearrange to get sec x = 2, which means cos x = 1/2. The solutions are x = π/3, 5π/3.