JEE Maths›Inverse Trigonometric Functions›Visual Solution
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Question
If α=sin⁡−1(513)\alpha = \sin^{-1}\left(\dfrac{5}{13}\right) and β=cos⁡−1(35)\beta = \cos^{-1}\left(\dfrac{3}{5}\right), then sin⁡(α+β)\sin(\alpha + \beta) equals:
Solution Path
Right triangle method: extract all trig ratios from the inverse trig values. Apply sine addition formula. sin(alpha+beta) = 63/65 = 0.969.
01Question Setup
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Given α=sin⁡−1(5/13)\alpha = \sin^{-1}(5/13) and β=cos⁡−1(3/5)\beta = \cos^{-1}(3/5), find sin⁡(α+β)\sin(\alpha + \beta).
sin⁡(α+β)=  ?\sin(\alpha+\beta) = \;?
02Draw Right Triangles
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From α\alpha: 5-12-13 triangle gives sin⁡α=5/13\sin\alpha = 5/13, cos⁡α=12/13\cos\alpha = 12/13. From β\beta: 3-4-5 triangle gives cos⁡β=3/5\cos\beta = 3/5, sin⁡β=4/5\sin\beta = 4/5.
sin⁡α=513,  cos⁡α=1213,  sin⁡β=45,  cos⁡β=35\sin\alpha = \frac{5}{13},\; \cos\alpha = \frac{12}{13},\; \sin\beta = \frac{4}{5},\; \cos\beta = \frac{3}{5}
03Apply Addition FormulaKEY INSIGHT
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sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β=513⋅35+1213⋅45=15+4865\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta = \frac{5}{13} \cdot \frac{3}{5} + \frac{12}{13} \cdot \frac{4}{5} = \frac{15+48}{65}.
sin⁡(α+β)=6365\sin(\alpha+\beta) = \dfrac{63}{65}
04Final Answer
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63/65≈0.96963/65 \approx 0.969. Sanity check: the value is between 0 and 1 as expected for a sine value.
0.969\boxed{0.969}
Concepts from this question1 concepts unlocked
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Composition and Right Triangle Method

STANDARD

To evaluate sin(cos inverse(x)), draw a right triangle where cos(theta) = x. Then the adjacent side is x, hypotenuse is 1, and the opposite side is sqrt(1-x^2). So sin(theta) = sqrt(1-x^2). This method works for any composition: tan(sin inverse(x)) = x/sqrt(1-x^2), cos(tan inverse(x)) = 1/sqrt(1+x^2), etc.

sin⁡(cos⁡−1x)=1−x2,tan⁡(sin⁡−1x)=x1−x2,cos⁡(tan⁡−1x)=11+x2\sin(\cos^{-1}x) = \sqrt{1-x^2}, \quad \tan(\sin^{-1}x) = \frac{x}{\sqrt{1-x^2}}, \quad \cos(\tan^{-1}x) = \frac{1}{\sqrt{1+x^2}}

This is the go-to technique for evaluating any composition of trig with inverse trig. JEE tests this in MCQs where the answer choices are algebraic expressions. The right triangle method is faster than using identities.

Evaluate sin(tan inverse x)Simplify nested inverse trigFind trig ratios from inverse trig values
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