Evaluate tan(sin^(-1)(3/5)).

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Question: Evaluate tan(sin^(-1)(3/5)).

Options:

Correct Answer: 4/3

Solution:

Let θ = sin^(-1)(3/5). Then sin(θ) = 3/5 and using the Pythagorean theorem, cos(θ) = 4/5. Therefore, tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4.

Evaluate tan(sin^(-1)(3/5)).

Evaluate tan(sin^(-1)(3/5)).

Correct Answer: 3/4

Step 1: Let θ = sin^(-1)(3/5). This means that sin(θ) = 3/5.
Step 2: To find cos(θ), use the Pythagorean theorem. We know that sin^2(θ) + cos^2(θ) = 1.
Step 3: Calculate sin^2(θ): (3/5)^2 = 9/25.
Step 4: Substitute sin^2(θ) into the Pythagorean theorem: 9/25 + cos^2(θ) = 1.
Step 5: Solve for cos^2(θ): cos^2(θ) = 1 - 9/25 = 16/25.
Step 6: Take the square root to find cos(θ): cos(θ) = 4/5 (we take the positive root since θ is in the first quadrant).
Step 7: Now, find tan(θ) using the formula tan(θ) = sin(θ) / cos(θ).
Step 8: Substitute the values: tan(θ) = (3/5) / (4/5).
Step 9: Simplify the fraction: tan(θ) = 3/4.

Inverse Trigonometric Functions – Understanding how to evaluate inverse sine functions and their corresponding angles.
Pythagorean Theorem in Trigonometry – Applying the Pythagorean theorem to find the cosine of an angle when the sine is known.
Trigonometric Ratios – Calculating tangent as the ratio of sine to cosine.