Express tan θ in terms of sin θ.

By BYJU'S Exam Prep

Updated on: September 25th, 2023

We can express tan θ in terms of sin θ as tan θ = Sin θ / cos θ. In right triangle trigonometry, for acute angles, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This definition holds true in the second and fourth quadrants, where the sine is positive and the cosine is negative.

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Tan θ in terms of sin θ

Solution:

Using the trigonometric ratios and identities, we can solve the given problem in the following way:

The tangent of an angle θ is defined as the ratio of the sine of θ to the cosine of θ. This can be expressed as tan θ = sin θ / cos θ….(1)

According to the trigonometric identity sin²θ + cos²θ = 1

So, cos θ = √(1 – sin²θ).

By substituting the value of cos θ in equation (1), we get tan θ = sin θ / √(1 – sin²θ).

Therefore, the given problem can be solved by expressing the tan θ = sin θ / √(1 – sin²θ).

Summary:

Express tan θ in terms of sin θ.

Tan θ can be expressed in terms of sin θ as tan θ = sin θ / √(1 – sin²θ). The tangent function will be negative in the second and fourth quadrants because the sine is positive (y-coordinate is positive) while the cosine is negative (x-coordinate is negative).

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