Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\frac{\cos 40^{\circ}}{\sin 50^{\circ}}$$

Short answer

The exact value is 1.

Step-by-step solution

Identify Complementary Angles

Recognize that the angles involved are complementary. Complementary angles add up to 90 degrees. Here, we have 40 degrees and 50 degrees, which are complementary since 40° + 50° = 90°.

Apply Complementary Angle Theorem

Use the Complementary Angle Theorem which states that \( \text{sin}(90^{\circ} - \theta) = \text{cos}(\theta) \). Hence, \( \text{sin}(50^{\circ}) = \text{cos}(40^{\circ}) \).

Simplify the Expression

Substitute \( \text{cos}(40^{\circ}) \) for \( \text{sin}(50^{\circ}) \) in the original expression. Therefore, \( \frac{\text{cos}(40^{\circ})}{\text{sin}(50^{\circ})} = \frac{\text{cos}(40^{\circ})}{\text{cos}(40^{\circ})} \).

Calculate the Final Value

Since \( \frac{\text{cos}(40^{\circ})}{\text{cos}(40^{\circ})} = 1 \), the expression simplifies to 1.

Key concepts

Fundamental Identities

In trigonometry, fundamental identities form the backbone of many calculations. These identities are universally true for all angles and can help simplify complex problems.
Some key fundamental identities include:

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

The simplicity offered by these identities can make solving trigonometric problems much easier. When combined with angle relationships, such as complementary angles, they become even more powerful.

Complementary Angles

Complementary angles are pairs of angles that add up to 90 degrees. For example, 40° and 50° are complementary since 40° + 50° = 90°.
One useful theorem related to complementary angles is the Complementary Angle Theorem. It states:
\[ \text{sin}(90^{\text{circ}} - \theta) = \text{cos}(\theta) \]
This theorem shows the relationship between the sine and cosine of complementary angles. When working with complementary angles in a problem, always check if you can apply this theorem. It simplifies calculations immensely.
In our exercise, recognizing that 40° and 50° are complementary allowed us to use this theorem to simplify the expression. We substituted \[ \text{cos}(40^{\text{circ}}) = \text{sin}(50^{\text{circ}}) \]
making the expression \[ \frac{\text{cos}(40^{\text{circ}})}{\text{sin}(50^{\text{circ}})} = 1 \].

Trigonometric Ratios

Trigonometric ratios are relationships between the sides of a right triangle and its angles. The primary trigonometric ratios are sine, cosine, and tangent. These ratios are defined as:

• Sine (\text{sin}): Opposite side / Hypotenuse
• Cosine (\text{cos}): Adjacent side / Hypotenuse
• Tangent (\text{tan}): Opposite side / Adjacent side

Using these definitions, you can determine the trigonometric value for any angle in a right triangle.
For complementary angles, these ratios have an interesting relation. Specifically:
\[ \text{sin}(90^{\text{circ}} - \theta) = \text{cos}(\theta) \] and \[ \text{cos}(90^{\text{circ}} - \theta) = \text{sin}(\theta) \]
Therefore, understanding the trigonometric ratios and how they transform under complementary angles is crucial. It aids in simplifying and solving many trigonometric identities and problems quickly and efficiently.