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Question: If \(\operatorname{Sin}A=\dfrac{1}{3}\) and \(A+B=90\text{ degree}\) then find \(\cos B\)...

If Sin⁡A=13\operatorname{Sin}A=\dfrac{1}{3} and A+B=90 degreeA+B=90\text{ degree} then find cos⁡B\cos B

Explanation

Solution

We solve this question by using basic trigonometric formulae. We are required to find the value of cos⁡B\cos B and this can be calculated using the relation sin⁡(90∘−θ)=cos⁡(θ).\sin \left( 90{}^\circ -\theta \right)=\cos \left( \theta \right). This relation is known as the concept of complementary angles. Using this relation along with the given information in the question, we find the value of cos⁡B.\cos B.

Complete step by step solution:
In order to answer this question, let us note down the given data first. It is given that
⇒sin⁡A=13…(1)\Rightarrow \sin A=\dfrac{1}{3}\ldots \left( 1 \right)
It is also given that A+B=90∘.A+B=90{}^\circ . We need to find the value of cos⁡B\cos B and this can be done by substituting in the equation sin⁡A=13\sin A=\dfrac{1}{3} in terms of B. So, by representing angle A in terms of B, we can obtain the solution to the above question. Using the second equation,
⇒A+B=90∘\Rightarrow A+B=90{}^\circ
The above equation represents the concept of complementary angles. Two angles are said to be complementary angles if the sum of the two angles is 90∘.90{}^\circ . From this equation, we subtract both sides by B.
⇒A=90∘−B\Rightarrow A=90{}^\circ -B
We now have the angle A represented in terms of B. We substitute this value of A in the equation 1.
⇒sin⁡(90∘−B)=13\Rightarrow \sin \left( 90{}^\circ -B \right)=\dfrac{1}{3}
We use the basic trigonometric relation sin⁡(90∘−θ)=cos⁡θ.\sin \left( 90{}^\circ -\theta \right)=\cos \theta . Using this formula for the above equation,
⇒cos⁡B=13\Rightarrow \cos B=\dfrac{1}{3}
Hence, we have obtained the value of cos⁡B\cos B which is given as 13.\dfrac{1}{3}.

Note: We need to know the basic trigonometric relations and conversion from sine to cosine functions. It is important to note the conversion from sine to cosine is given by sin⁡(90∘−θ)=cos⁡θ\sin \left( 90{}^\circ -\theta \right)=\cos \theta and vice versa equation is given by cos⁡(90∘−θ)=sin⁡θ.\cos \left( 90{}^\circ -\theta \right)=\sin \theta . This concept is called the complementary angles concept. We can use this to solve many mathematical questions.