Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\cos 35^{\circ} \sin 55^{\circ}+\cos 55^{\circ} \sin 35^{\circ}$$

Short answer

\( 1 \)

Step-by-step solution

Identify the given trigonometric expression

The given expression is \( \cos 35^{\circ} \sin 55^{\circ} + \cos 55^{\circ} \sin 35^{\circ} \).

Recognize the sum-to-product identity

The sum-to-product identity for the given form is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).

Apply the identity

By comparing the given expression \( \cos 35^{\circ} \sin 55^{\circ} + \cos 55^{\circ} \sin 35^{\circ} \) to the sum-to-product formula, we can see that it matches \( \sin(A + B) \). Therefore, the expression becomes \( \sin(35^{\circ} + 55^{\circ}) \).

Simplify the angle

Add the angles inside the sine function: \( 35^{\circ} + 55^{\circ} = 90^{\circ} \).

Find the value of the sine function

Recall that \( \sin 90^{\circ} = 1 \). Therefore, \( \sin(90^{\circ}) = 1 \).

Key concepts

Sum-to-Product Identities

Sum-to-product identities are a set of trigonometric identities that transform sums or differences of sine and cosine into products. These identities help simplify complex trigonometric expressions. For example, the sum-to-product identity for sine is:
\[ \text{sin}(A + B) = \text{sin} A \text{cos} B + \text{cos} A \text{sin} B \]
This particular identity was used to solve our original problem.
By recognizing that the expression matched the sum-to-product form, we could rewrite it as a single sine term.

Sine Function

The sine function, denoted as \( \text{sin}(x) \), is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. In our example, we used the sine function in the final step:

• Adding the angles, \(\text{sin}(35^{\text{°}} + 55^{\text{°}})\), resulted in \(\text{sin}(90^{\text{°}})\).
• Knowing the value of \(\text{sin}(90^{\text{°}}) = 1\) allowed us to simplify the expression completely.

It's essential to remember the basic values of sine for common angles, such as \( \text{sin}(0^{\text{°}}) = 0 \), \( \text{sin}(30^{\text{°}}) = \frac{1}{2} \), \( \text{sin}(45^{\text{°}}) = \frac{\text{sqrt}(2)}{2} \), \( \text{sin}(60^{\text{°}}) = \frac{\text{sqrt}(3)}{2} \), and \( \text{sin}(90^{\text{°}}) = 1 \).

Complementary Angle Theorem

The Complementary Angle Theorem states that the sine of an angle is equal to the cosine of its complement. Two angles are complementary if their sum is \( 90^{\text{°}} \). For instance,

• In our example, \(\text{sin}(35^{\text{°}})\) is equal to \(\text{cos}(55^{\text{°}})\) since \(35^{\text{°}}\) and \(55^{\text{°}}\) are complementary angles.
• Therefore,\(\text{cos}(35^{\text{°}})\) and \(\text{sin}(55^{\text{°}})\) are also complementary.

This theorem simplifies calculations by allowing the interchangeability of sine and cosine when dealing with complementary angles.
In the original problem, recognizing that \(35^{\text{°}} + 55^{\text{°}} = 90^{\text{°}}\) confirms how sine and cosine functions relate in such expressions.