Problem 20 - Precalculus : Building Concepts and Connections

Question

Use the given value of a trigonometric function of \(\theta\) to find the values of the other five trigonometric functions. Assume \(\theta\) is an acute angle. $$\cos \theta=0.8$$

Step-by-Step Solution

Answer

The six trigonometric functions of \(\theta\) are: \(\sin \theta = 0.6\), \(\cos \theta = 0.8\), \(\tan \theta = 0.75\), \(\csc \theta = 1.66667\), \(\sec \theta = 1.25\), \(\cot \theta = 1.33333\).

Step 1: Compute \(\sin \theta\)

Use the Pythagorean Identity \(\sin^2 \theta + \cos^2 \theta = 1\) to compute \(\sin \theta\). Substitute \(\cos \theta = 0.8\) into the identity, then solve for \(\sin \theta\) getting \(\sin \theta = \sqrt{1 - (0.8)^2} = 0.6\). As \(\theta\) is an acute angle, \(\sin \theta\) is positive.

Step 2: Compute \(\tan \theta\)

Find \(\tan \theta\) using the relationship \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the given \(\cos \theta = 0.8\) and computed \(\sin \theta = 0.6\) into the equation to get \(\tan \theta = 0.6 / 0.8 = 0.75\)

Step 3: Compute reciprocals

Now, to find \(\csc \theta\), \(\sec \theta\), and \(\cot \theta\), merely take the reciprocals of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) respectively. These yield: \(\csc \theta = 1 / \sin \theta = 1 / 0.6 = 1.66667\), \(\sec \theta = 1 / \cos \theta = 1 / 0.8 = 1.25\), and \(\cot \theta = 1 / \tan \theta = 1 / 0.75 = 1.33333\).

Step 4: Closing Remarks

Thus, the six trigonometric functions of the angle \(\theta\) with \(\cos \theta = 0.8\) and \(\theta\) being an acute angle are: \(\sin \theta = 0.6\), \(\cos \theta = 0.8\), \(\tan \theta = 0.75\), \(\csc \theta = 1.66667\), \(\sec \theta = 1.25\), and \(\cot \theta = 1.33333\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relation in trigonometry that links the squares of the sine and cosine functions of an angle. It is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is immensely useful, particularly for finding one trigonometric function when another is known.
In our specific exercise where \( \cos \theta = 0.8 \), we applied the Pythagorean Identity to find \( \sin \theta \). By substituting into the identity, we get \( \sin^2 \theta + 0.8^2 = 1 \).
This simplifies to \( \sin^2 \theta = 1 - 0.64 = 0.36 \). Finally, taking the square root, we find \( \sin \theta = 0.6 \).

**Key Points:**

• Pythagorean Identity connects sine and cosine.
• Useful for finding unknown trigonometric values.
• Always ensure values are positive in the appropriate quadrant for acute angles.

Reciprocal Trigonometric Functions

Reciprocal Trigonometric Functions are secondary trigonometric functions obtained by taking the reciprocals of the primary trigonometric functions. These are

• Cosecant, \( \csc \theta = \frac{1}{\sin \theta} \)
• Secant, \( \sec \theta = \frac{1}{\cos \theta} \)
• Cotangent, \( \cot \theta = \frac{1}{\tan \theta} \)

In the exercise, knowing \( \sin \theta = 0.6 \) and \( \cos \theta = 0.8 \), it becomes easy to find these reciprocal functions.
For \( \csc \theta \), we calculate \( \frac{1}{0.6} \), resulting in approximately 1.66667. Similarly, \( \sec \theta = \frac{1}{0.8} = 1.25 \), and \( \cot \theta = \frac{1}{0.75} = 1.33333 \).
This step is crucial for various applications in trigonometry, particularly in solving triangles and understanding function behaviors.

Acute Angle Trigonometry

Acute Angle Trigonometry deals with angles less than 90 degrees. In this context, all six trigonometric functions are positive. This fundamental property helps in simplifying calculations by focusing on their inherent positivity in the relevant quadrant.
In our example, \(\theta\) is an acute angle, which means

• Both \(\sin \theta\) and \(\cos \theta\) are positive.
• Similarly, \(\tan \theta\), along with its reciprocal \(\cot \theta\), are positive.

When calculating trigonometric values for acute angles, always assume positivity once the angle's location within the first quadrant is confirmed.

This step makes the initial exercise easier because confirming \(\cos \theta = 0.8\) directly points to a positive \(\sin \theta\), ensuring all calculated values align with our acute angle setup.