Problem 45 - Precalculus Essentials

Question

Find the exact value of each expression. Do not use a calculator. $$1+\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(2\)

Step 1: Recognize the pattern

This is a trigonometry problem dealing with the values of sinus function at two different angles. Notice that the two angles add up to 90 degrees, which can be taken advantage of using the complementary angles property.

Step 2: Apply the identity of complementary angles

According to the law of complementary angles for sinus function: \(\sin^{2}(\theta) + \sin^{2}(90-\theta) = 1\). Apply this property to the expression, we have \(1+\sin^{2}(40) + \sin^{2}(50) = 1 + 1 = 2\)

Step 3: Write down the result

After applying the trigonometric law of complementary angles, we obtain the final result as: \(2\)