Introduction / Context:
This trigonometry question asks you to convert from one trigonometric ratio to another using a right triangle model. You are given tan θ and asked to find sin θ for an acute angle θ. Instead of working with approximate decimal values directly, you can interpret the ratio 4/3 as side lengths in a right triangle and then apply the Pythagoras theorem to find all sides and the required sine value.
Given Data / Assumptions:
- tan θ = 4/3 for an acute angle θ.
- tan θ = opposite side / adjacent side.
- sin θ = opposite side / hypotenuse.
- Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2.
- θ is acute, so all basic trigonometric values are positive.
Concept / Approach:
Interpret tan θ = 4/3 by choosing a right triangle where the side opposite θ is 4 units and the side adjacent to θ is 3 units. This choice fits the given ratio and is convenient. Then apply Pythagoras theorem to calculate the hypotenuse. Once you have all three side lengths, compute sin θ as opposite divided by hypotenuse. Finally, convert the fraction to a decimal to compare with the options if needed.
Step-by-Step Solution:
Let the opposite side to θ be 4 and the adjacent side be 3, so tan θ = 4/3.
Apply Pythagoras theorem: hypotenuse^2 = 4^2 + 3^2 = 16 + 9 = 25.
So the hypotenuse is 5, since 5^2 = 25.
Now sin θ = opposite / hypotenuse = 4 / 5.
Convert 4/5 to a decimal: 4/5 = 0.8.
Verification / Alternative check:
In the well known 3–4–5 Pythagorean triangle, if the sides opposite and adjacent to the angle are 4 and 3 respectively, tan θ = 4/3, sin θ = 4/5, and cos θ = 3/5. All these satisfy the identity sin^2 θ + cos^2 θ = 1, since (4/5)^2 + (3/5)^2 = 16/25 + 9/25 = 25/25 = 1. This confirms that sin θ = 4/5, which is 0.8 in decimal form.
Why Other Options Are Wrong:
Option b (1.25) is greater than 1 and therefore cannot be a sine value, since sine values are bounded between -1 and 1. Option c (4/3) is tan θ itself, not sin θ. Option d (3/4) is the cosine of θ, which is adjacent over hypotenuse in this triangle. Option e (1/2) is often associated with sin 30 degrees but does not match the triangle representing tan θ = 4/3.
Common Pitfalls:
Learners sometimes mistakenly treat 4/3 as opposite/hypotenuse or adjacent/hypotenuse instead of opposite/adjacent, which leads to inconsistent side lengths. Another common error is to attempt to approximate values with a calculator rather than using the exact Pythagorean triple. Remembering the 3–4–5 triangle and the definitions of tan, sin, and cos makes these conversions much easier.
Final Answer:
The exact value of sin θ for tan θ = 4/3 is 4/5, which equals
0.8 in decimal form.
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