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Chapter 5: Problem 65

Use the cofunction identities to evaluate the expression without using a calculator. $$\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ}$$

Short Answer

Expert verified
The answer is 1.

Step by step solution

01

Identify the Problem

You are asked to evaluate \( \sin^2 25掳 + \sin^2 65掳 \) without using a calculator. The most important aspect is recognizing that 25掳 and 65掳 are complementary angles (they add up to 90掳), so cofunction identities can be applied.
02

Apply Cofunction Identity

According to the cofunction identity \( \sin{(\Theta)} = \cos{(90^\circ - \Theta)} \), you can replace \( \sin^2 65掳 \) with \( \cos^2 25掳 \) because \( 90^\circ - 25^\circ = 65^\circ \). So, \( \sin^2 25掳 + \sin^2 65掳 \) becomes \( \sin^2 25掳 + \cos^2 25掳 \).
03

Use Pythagorean Identity

Now, you apply the Pythagorean Identity which states \( \sin^2 \Theta + \cos^2 \Theta = 1 \). In this case, \( \Theta = 25^{\circ} \), hence \( \sin^2 25掳 + \cos^2 25掳 = 1 \).

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