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Difficulty: MediumTrigonometric Ratios and Identities

In right triangle ABCABC, the measure of angle CC is 90∘90^\circ. If cos⁡(A)=3sin⁡(A)\cos(A) = 3\sin(A), what is the value of tan⁡(B)\tan(B)?

Answer: 3

Answer

The correct answer is 33.
Because angle CC is 90∘90^\circ in right triangle ABCABC, the acute angles AA and BB are complementary (A+B=90∘A + B = 90^\circ). By the co-function identities, sin⁡(B)=cos⁡(A)\sin(B) = \cos(A) and cos⁡(B)=sin⁡(A)\cos(B) = \sin(A). The tangent of BB is defined as tan⁡(B)=sin⁡(B)cos⁡(B)\tan(B) = \frac{\sin(B)}{\cos(B)}. Substituting the co-function identities gives tan⁡(B)=cos⁡(A)sin⁡(A)\tan(B) = \frac{\cos(A)}{\sin(A)}. Since we are given that cos⁡(A)=3sin⁡(A)\cos(A) = 3\sin(A), we substitute this expression into the numerator to get tan⁡(B)=3sin⁡(A)sin⁡(A)=3\tan(B) = \frac{3\sin(A)}{\sin(A)} = 3.

Step-by-Step Solution

1
Determine the relationship between the acute angles in right triangle ABCABC.
sin⁡(B)=cos⁡(A)\sin(B) = \cos(A) and cos⁡(B)=sin⁡(A)\cos(B) = \sin(A)
Since angle CC is 90∘90^\circ, the other two angles AA and BB must sum to 90∘90^\circ (they are complementary angles).
2
Express tan⁡(B)\tan(B) in terms of the trigonometric ratios of angle AA.
tan⁡(B)=cos⁡(A)sin⁡(A)\tan(B) = \frac{\cos(A)}{\sin(A)}
By definition, the tangent of angle BB is the ratio of its sine to its cosine, which yields cos⁡(A)sin⁡(A)\frac{\cos(A)}{\sin(A)} after substituting the complementary angle relations.
3
Substitute the given relation cos⁡(A)=3sin⁡(A)\cos(A) = 3\sin(A) into the expression for tan⁡(B)\tan(B).
tan⁡(B)=3sin⁡(A)sin⁡(A)\tan(B) = \frac{3\sin(A)}{\sin(A)}
Substituting the value of cos⁡(A)\cos(A) allows us to simplify the fraction by expressing both terms with sin⁡(A)\sin(A).
4
Simplify the fraction to get the final numerical value.
tan⁡(B)=3\tan(B) = 3
The term sin⁡(A)\sin(A) cancels out from the numerator and denominator since AA is an acute angle and sin⁡(A)≠0\sin(A) \neq 0.

Key Concept

Co-function identities and trigonometric definitions in a right triangle.

Alternative Method

Alternatively, construct a right triangle where the side opposite to angle AA has length 11. Since cos⁡(A)=3sin⁡(A)\cos(A) = 3\sin(A), the ratio of the adjacent side to the hypotenuse is 33 times the ratio of the opposite side to the hypotenuse, meaning the side adjacent to angle AA must have length 33. Because angle BB is the complement of angle AA, the side opposite to angle BB is the side adjacent to angle AA (which is 33), and the side adjacent to angle BB is the side opposite to angle AA (which is 11). Therefore, tan⁡(B)=oppositeadjacent=31=3\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{1} = 3.
Estimated Time:1m 30s
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Topics

Trigonometric Ratios and Identities

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