Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\sin 38^{\circ}-\cos 52^{\circ}$$

Short answer

0

Step-by-step solution

Recall the Complementary Angle Theorem

The Complementary Angle Theorem states that \( \theta + (90^\circ - \theta) = 90^\circ \). Therefore, \( \theta \) and \( 90^\circ - \theta \) are complementary angles. This implies that \( \sin(90^\circ - \theta) = \cos(\theta) \).

Apply the Complementary Angle Theorem

Recognize that \( 52^\circ = 90^\circ - 38^\circ \). Using the Complementary Angle Theorem, \( \sin(90^\circ - 38^\circ) = \cos(38^\circ) \). In other words, \( \cos(52^\circ) = \sin(38^\circ) \).

Substitute and Simplify

Given \( \sin(38^\circ) - \cos(52^\circ) \, \), substitute \( \cos(52^\circ) \) with \( \sin(38^\circ) \). The expression now becomes: \( \sin(38^\circ) - \sin(38^\circ) \).

Solve the Expression

After substituting, simplify the expression: \( \sin(38^\circ) - \sin(38^\circ) = 0 \).

Key concepts

Complementary Angle Theorem

The Complementary Angle Theorem is essential in understanding certain trigonometric relationships. Complementary angles are two angles whose measures add up to 90 degrees. According to this theorem, the sine of one angle is equal to the cosine of its complement. In mathematical terms, if you have an angle \(\theta\), then the complementary angle is \(90^\circ - \theta\). This gives us the identity: \( \sin(90^\circ - \theta) = \cos(\theta) \). This relationship helps to transform trigonometric expressions involving sine and cosine into simpler forms. For example, in our problem, recognizing that 52 degrees is the complement of 38 degrees (since 38 + 52 = 90), we can use this theorem to simplify our calculation for \( \sin(38^\circ) - \cos(52^\circ)\). This insight dramatically simplifies the solution process, turning potentially tricky problems into straightforward calculations.

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities include a set of equations that are always true for any angle. These identities are deeply rooted in the properties of right-angle triangles and the unit circle. The most common fundamental identities include:

• Pythagorean Identities: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal Identities: \( \sin(\theta) = \frac{1}{\csc(\theta)}, \cos(\theta) = \frac{1}{\sec(\theta)}\)
• Quotient Identities: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)

These identities are crucial when simplifying complex trigonometric expressions. In the exercise, we used the Complementary Angle Theorem, which is tied to these fundamental identities, to simplify \( \sin(38^\circ) - \cos(52^\circ)\). Recognizing and applying these identities will make solving trigonometric equations much more straightforward.

Sine and Cosine Relationships

The relationship between sine and cosine is a core concept in trigonometry. They are directly linked through the complementary angle theorem as well as the Pythagorean identity. This means that understanding one function can help in understanding the other. Here are some key relationships:

• Complementary Relationship: \( \sin(90^\circ - \theta) = \cos(\theta) \)
• Phase Shift: \( \sin(\theta) = \cos(90^\circ - \theta) \)
• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

In our exercise, we demonstrated this by identifying that \(\cos(52^\circ)\) equals \(\sin(38^\circ)\) due to the complementary angle theorem. This step showcases how interconnected these functions are. By leveraging these relationships, solving trigonometric functions becomes much simpler.