Chapter 9: Problem 5

Show \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\).

Short Answer

Expert verified

Question: Prove the following trigonometric identity: \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\) Answer: Using the co-function identity formula \(\sin \left(\frac{\pi}{2} - x\right) = \cos(x)\), we can plug in our expression and confirm the result as follows: \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos(\theta)\). This verifies that \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\).

Step by step solution

Recall the co-function identity formula

In this case, we will be using the following co-function identity formula: \(\sin \left(\frac{\pi}{2} - x\right) = \cos(x)\) The formula demonstrates that the sine of the complementary angle to x (90 degrees, or pi/2 radians, minus x) is equal to the cosine of x.

Apply the co-function identity to our expression

Now that we know the co-function identity formula, let's apply it to the expression we have to prove. We are trying to show that \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\). We can just plug in the formula for the sine of the complementary angle: \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos(\theta)\)

Confirm the result

Based on the co-function identity formula, our expression holds true: \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos \theta\) That demonstrates that \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\). The co-function identity allowed us to connect the sine and cosine functions and prove the given identity.

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Show \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\).

Short Answer

Step by step solution

Recall the co-function identity formula

Apply the co-function identity to our expression

Confirm the result

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