When mA + mB = 90°, what relationship is formed by tan A and tan B? Select all that apply: (1 Point) 13.1 Tangent Ratio Problem – HOT Question tan A tan B tan A tan B = | tan² A tan² B = [ (tan A) (tan B) = 1 (tan A) (tan B) = -1
The Correct Answer and Explanation is:
Given that m∠A + m∠B = 90°, we are told that the angles A and B are complementary. This forms the basis of a fundamental trigonometric identity.
Correct Answer:
(tan A)(tan B) = 1
Explanation
When two angles are complementary, their measures add up to 90°. In this case, m∠A + m∠B = 90°, so we can write: ∠B=90∘−∠A\angle B = 90^\circ – \angle A
In trigonometry, the tangent of an angle in a right triangle is defined as: tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
There’s also a very important identity that relates the tangent of complementary angles: tan(A)=cot(B)if A+B=90∘\tan(A) = \cot(B) \quad \text{if } A + B = 90^\circ
But since: cot(B)=1tan(B),\cot(B) = \frac{1}{\tan(B)},
we get: tan(A)=1tan(B)⇒tan(A)⋅tan(B)=1\tan(A) = \frac{1}{\tan(B)} \Rightarrow \tan(A) \cdot \tan(B) = 1
This is the key relationship that holds when two angles are complementary.
Let’s test this with an example:
- Let A=30∘A = 30^\circ → then B=60∘B = 60^\circ
- tan(30∘)≈0.577\tan(30^\circ) \approx 0.577
- tan(60∘)≈1.732\tan(60^\circ) \approx 1.732
- Product: 0.577⋅1.732≈10.577 \cdot 1.732 \approx 1
This confirms the identity.
Why Other Options Are Incorrect:
- tan(A)+tan(B)\tan(A) + \tan(B): No specific identity relates this directly to complementary angles.
- tan2(A)tan2(B)\tan^2(A) \tan^2(B): This is the square of the correct identity, which equals 12=11^2 = 1, but it’s not the basic or most direct relationship.
- (tanA)(tanB)=−1(\tan A)(\tan B) = -1: This is incorrect. That identity applies in perpendicular lines in coordinate geometry, not complementary angles.
Conclusion:
For two complementary angles, the product of their tangents is always 1. Thus, the correct relationship is: (tanA)(tanB)=1(\tan A)(\tan B) = 1
