Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\sin 80^{\circ} \csc 80^{\circ}$$

Short answer

1

Step-by-step solution

Understand the Problem

The exercise asks to find the exact value of the expression involving \( \sin 80^{\circ} \) and \( \csc 80^{\circ} \). Note that \( \csc \theta \) is the reciprocal of \( \sin \theta \), meaning \( \csc \theta = \frac{1}{\sin \theta} \).

Substitute the Reciprocal Identity

Express \( \csc 80^{\circ} \) in terms of \( \sin 80^{\circ} \). The identity states that \( \csc 80^{\circ} = \frac{1}{\sin 80^{\circ}} \).

Simplify the Expression

Substitute \( \csc 80^{\circ} \) into the original expression: \( \sin 80^{\circ} \cdot \csc 80^{\circ} \) becomes \( \sin 80^{\circ} \cdot \frac{1}{\sin 80^{\circ}} \).

Simplify Further

Using the identity from the previous step, the expression \( \sin 80^{\circ} \cdot \frac{1}{\sin 80^{\circ}} \) simplifies to \( 1 \).

Key concepts

Complementary Angle Theorem

The Complementary Angle Theorem is a useful tool in trigonometry. It states that the sine of an angle is equal to the cosine of its complementary angle. Simply put, if two angles add up to 90 degrees, one is the complement of the other. For example:
\begin{enumerate}

\(\text{sin} (90^\text{°} - \theta) = \text{cos} (\theta)\)

\(\text{cos} (90^\text{°} - \theta) = \text{sin} (\theta)\)

This theorem is particularly handy when simplifying trigonometric expressions that combine sine and cosine. It gives a quick conversion between these functions, making complex problems simpler. In the provided exercise, understanding this relationship didn’t directly change the solution, but it's crucial knowledge for other trigonometric tasks.

Reciprocal Identity

Reciprocal Identities make it easier to handle trigonometric expressions. They show how the six trigonometric functions relate to each other through reciprocals. For a quick reminder:

\(\text{csc} (\theta) = \frac{1}{\text{sin} (\theta)}\)
\(\text{sec} (\theta) = \frac{1}{\text{cos} (\theta)}\)
\(\text{cot} (\theta) = \frac{1}{\text{tan} (\theta)}\)

In the problem, we used the reciprocal identity \(\text{csc} (80^\text{°}) = \frac{1}{\text{sin} (80^\text{°})}\) to simplify the expression. Understanding this identity transforms a seemingly complex expression into a straightforward calculation. This knowledge is fundamental in many trigonometric problems you will encounter.

Exact Trigonometric Values

Exact Trigonometric Values refer to precise values of trigonometric functions for specific angles, commonly used in math problems to avoid approximation errors. These values are especially useful for angles like 0°, 30°, 45°, 60°, and 90°. These exact values include:

• \(\text{sin} (45^\text{°}) = \frac{\text{√2}}{2}\)
• \(\text{cos} (45^\text{°}) = \frac{\text{√2}}{2}\)
• \(\text{tan} (45^\text{°}) = 1\)
• \(\text{sin} (30^\text{°}) = \frac{1}{2}\)
• \(\text{cos} (30^\text{°}) = \frac{\text{√3}}{2}\)
• \(\text{tan} (30^\text{°}) = \frac{\text{√3}}{3}\)
• \(\text{sin} (60^\text{°}) = \frac{\text{√3}}{2}\)
• \(\text{cos} (60^\text{°}) = \frac{1}{2}\)
• \(\text{tan} (60^\text{°}) = \text{√3}\)

Memorizing these values can save you time and effort on exams. While the given problem didn’t require these specific values, having them in your toolkit is invaluable for many trigonometric problems.