Question

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\cos \theta=\frac{\sqrt{2}}{2}$$

Short answer

\sin \theta = \frac{\sqrt{2}}{2}, \tan \theta = 1, \sec \theta = \sqrt{2}, \csc \theta = \sqrt{2}, \cot \theta = 1.

Step-by-step solution

Determine \sin \theta using the Pythagorean identity

Use the Pythagorean identity \sin^2 \theta + \cos^2 \theta = 1. Since \cos \theta = \frac{\sqrt{2}}{2}, we have: \[ \sin^2 \theta + \(\frac{\sqrt{2}}{2}\)^2 = 1 \sin^2 \theta + \frac{1}{2} = 1 \sin^2 \theta = 1 - \frac{1}{2} \sin^2 \theta = \frac{1}{2} \sin \theta = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}\rightarrow \text{( Taking the positive value since \theta is an acute angle.)} \]

Determine \tan \theta

To find \tan \theta use \tan \theta = \frac{\sin \theta}{\cos \theta}. \[ \tan \theta = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \]

Determine \sec \theta

Use the reciprocal identity \sec \theta = \frac{1}{\cos \theta}. \[ \sec \theta = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

Determine \csc \theta

Use the reciprocal identity \csc \theta = \frac{1}{\sin \theta}. \[ \csc \theta = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

Determine \cot \theta

Use the reciprocal identity \cot \theta = \frac{1}{\tan \theta}. \[ \cot \theta = \frac{1}{1} = 1 \]

Key concepts

Pythagorean identity

The Pythagorean identity is a fundamental trigonometric identity that relates the squares of the sine and cosine functions. It states: \[ \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \] This identity is derived from the Pythagorean theorem applied to a right-angled triangle. In the context of an acute angle \(\theta\), knowing \(\text{cos }\theta = \frac{\sqrt{2}}{2}\) allows us to find \(\text{sin }\theta\) by rearranging the identity: \[ \text{sin}^2 \theta = 1 - \text{cos}^2 \theta \] Substituting the given cosine value, we get: \[ \text{sin}^2 \theta = 1 - \left( \frac{\sqrt{2}}{2} \right)^2 = 1 - \frac{1}{2} = \frac{1}{2} \] Hence, \(\text{sin }\theta = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}\), taking the positive value since \(\theta\) is acute.

Reciprocal identities

Reciprocal trigonometric identities provide a relationship between pairs of trigonometric functions, making it easier to switch between different functions. The key reciprocal identities are:

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

For example, knowing \(\text{cos }\theta = \frac{\sqrt{2}}{2}\), we can find \(\text{sec }\theta\): \[ \text{sec }\theta = \frac{1}{\text{cos }\theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \] Similarly, for \(\text{sin }\theta = \frac{\sqrt{2}}{2}\), \(\text{csc }\theta\) is: \[ \text{csc }\theta = \frac{1}{\text{sin }\theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \] And for \(\text{tan }\theta = 1\), \(\text{cot }\theta\) is simply: \[ \text{cot }\theta = \frac{1}{\text{tan }\theta} = \frac{1}{1} = 1 \] Using these identities significantly simplifies computations involving trigonometric functions.

Acute angle trigonometric functions

An acute angle is an angle that measures less than 90 degrees. For trigonometric functions involving acute angles, all the trigonometric values are taken as positive. This is because acute angles are situated in the first quadrant of the unit circle, where sine, cosine, and tangent values are positive. In our problem, knowing that \(\theta\) is acute ensures that \(\text{sin }\theta = \frac{\sqrt{2}}{2}\) is positive, as opposed to negative. Let's summarize the exact values:

• \(\text{sin }\theta = \frac{\sqrt{2}}{2}\)
• \(\text{cos }\theta = \frac{\sqrt{2}}{2}\)
• \(\text{tan }\theta = 1\)
• \(\text{sec }\theta = \sqrt{2}\)
• \(\text{csc }\theta = \sqrt{2}\)
• \(\text{cot }\theta = 1\)

These values remain consistent for any acute angle, making it crucial to understand the properties of acute angles in trigonometry.

Exact trigonometric function values

Exact trigonometric values for specific angles (like 30°, 45°, 60°) are essential for solving many trigonometric problems without a calculator. In our example, the given value \(\text{cos }\theta = \frac{\sqrt{2}}{2}\) corresponds to \(\theta = 45^\text{o}\). This is because:
- For \(45^\text{o}\), \(\text{cos }\theta = \frac{\sqrt{2}}{2}\) and \(\text{sin }\theta = \frac{\sqrt{2}}{2}\).
These exact values help in determining other trigonometric functions:
- \(\text{tan }45^\text{o} = 1\)
- \(\text{sec }45^\text{o} = \sqrt{2}\)
- \(\text{csc }45^\text{o} = \sqrt{2}\)
- \(\text{cot }45^\text{o} = 1\)
Remembering these relationships allows you to quickly solve problems without having to recalculate trigonometric values each time. This makes trigonometry easier to work with in exams and practical applications.