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If tanθ=158, then the value of 1+sinθ1−sinθ is:
3/5
3/4
5/7
3/7
3/5
Since tan θ = 8/15, the hypotenuse is sqrt(8² + 15²) = 17. Thus, sin θ = 8/17 and cos θ = 15/17. The expression becomes sqrt((1 - 8/17)/(1 + 8/17)) = sqrt((9/17)/(25/17)) = sqrt(9/25) = 3/5.
tan θ = 8/15. We need to find the value of the expression sqrt((1 - sin θ) / (1 + sin θ)).
1+sinθ1−sinθ=cosθ1−sinθ=secθ−tanθ
Since tan θ = 8/15, the hypotenuse is sqrt(8² + 15²) = 17. Thus, sin θ = 8/17 and cos θ = 15/17. The expression becomes sqrt((1 - 8/17)/(1 + 8/17)) = sqrt((9/17)/(25/17)) = sqrt(9/25) = 3/5.
Students often struggle to simplify the nested radical or fail to recognize that the identity results in a rational fraction, choosing to calculate decimals instead.
Find the hypotenuse
Given tanθ=158, which is adjacentopposite. The hypotenuse h=82+152=64+225=289=17.
h=17
Calculate sin θ and cos θ
Using the triangle values, sinθ=178 and cosθ=1715.
sinθ=178,cosθ=1715
Substitute into the expression
The expression is 1+sinθ1−sinθ=1+8/171−8/17.
25/179/17=259=53
A is correct because the final calculated value of the expression is 3/5.
This type of simplification often appears in identity proofs; remember that 1+sinθ1−sinθ is a standard form that simplifies to secθ−tanθ.
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