Difficulty: Easy
Correct Answer: 0.8
Explanation:
Introduction / Context:
This trigonometry question checks your ability to convert between different trigonometric ratios. You are given tan θ and asked to find sin θ. By interpreting tan θ as a ratio of sides in a right triangle and using the Pythagorean theorem, you can determine all relevant side lengths and then compute sin θ directly.
Given Data / Assumptions:
Concept / Approach:
The definition tan θ = opposite / adjacent allows us to assign lengths to the opposite and adjacent sides of a right triangle. Once these two sides are known, we use the Pythagorean theorem to find the hypotenuse. Then sin θ = opposite / hypotenuse can be computed. This approach is straightforward and relies on the well known 3-4-5 right triangle pattern.
Step-by-Step Solution:
Verification / Alternative check:
Check that tan θ remains 4/3 with these side assignments. Using opposite 4 and adjacent 3, tan θ = 4 / 3 as given. Also, verify that 3-4-5 is a valid right triangle triple because 3^2 + 4^2 = 9 + 16 = 25 = 5^2. This confirms the triangle is consistent and that sin θ = 4/5 or 0.8 is reliable.
Why Other Options Are Wrong:
The value 1.25 equals 5/4 and would correspond to a sine greater than 1, which is impossible. The value 4/3 is tan θ itself, not sin θ. The value 3/4 corresponds to cos θ for this triangle, not sin θ. The value 5/4 is again greater than 1 and cannot be a sine value. Only 0.8, which equals 4/5, fits the correct ratio of opposite to hypotenuse.
Common Pitfalls:
Students sometimes confuse tan θ with sin θ and mistakenly think they are equal. Others may forget to compute the hypotenuse and directly assume sin θ is 4/3, which is not allowed because sine values must lie between −1 and 1. Carefully assigning sides and using the Pythagorean theorem avoids these errors.
Final Answer:
The value of sin θ, given tan θ = 4/3, is 0.8 (which is 4/5).
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