Problem 22 - 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. $$\cos \theta=0.7$$

Step-by-Step Solution

Answer

\(\sin \theta \approx 0.714\), \(\tan \theta \approx 1.02\), \(\csc \theta \approx 1.4\), \(\sec \theta \approx 1.43\), \(\cot \theta \approx 0.98\)

Step 1: Use Pythagorean identity to find \(\sin \theta\)

Use the relationship \(\sin^2 \theta = 1 - \cos^2 \theta\) to find \(\sin \theta\). Substituting \(\cos \theta = 0.7\) we get \(\sin^2 \theta= 1 - 0.7^2 = 1 - 0.49 = 0.51\), thus \(\sin \theta = \sqrt{0.51}\ \approx 0.714\). Since we know \(\theta\) is acute, \(\sin \theta\) is positive.

Step 2: Find value of \(\tan \theta\)

\(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting \(\sin \theta = 0.714\) and \(\cos \theta = 0.7\) we find that \(\tan \theta = \frac{0.714}{0.7} \approx 1.02\)

Step 3: Find the remaining trigonometric functions using principles of co-function

The co-functions can be calculated as \(\csc \theta = 1/\sin \theta\), \(\sec \theta = 1/\cos \theta\), and \(\cot \theta = 1/\tan \theta\). Substituting the known values gives \(\csc \theta = 1/0.714 \approx 1.4\), \(\sec \theta = 1/0.7 \approx 1.43\), and \(\cot \theta = 1/1.02 \approx 0.98\)

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry. It's derived from the Pythagorean theorem, which relates to the sides of a right triangle. This identity states: \[\sin^2 \theta + \cos^2 \theta = 1\]This relationship is extremely useful when one trigonometric function is given, as it allows you to find the others. In our example, knowing \(\cos \theta = 0.7\), we can use this identity to find \(\sin \theta\). By rearranging the formula, we get:

• \(\sin^2 \theta = 1 - \cos^2 \theta\)
• \(\sin^2 \theta = 1 - (0.7)^2\)
• \(\sin^2 \theta = 0.51\)
• \(\sin \theta = \sqrt{0.51} \approx 0.714\). Since \(\theta\) is acute, \(\sin \theta\) is positive.

Thus, the Pythagorean Identity simplifies the process of determining unknown trigonometric functions, provided you start with one known value.

Co-function Identities

Co-function identities connect trigonometric functions of complementary angles. This concept is centered around the notion that for any angle \(\theta\), the trigonometric function of its complement can be expressed in terms of co-functions. In simpler terms, these identities are:

• \(\sin(90^\circ - \theta) = \cos \theta\)
• \(\cos(90^\circ - \theta) = \sin \theta\)
• \(\tan(90^\circ - \theta) = \cot \theta\)
• \(\csc(90^\circ - \theta) = \sec \theta\)
• \(\sec(90^\circ - \theta) = \csc \theta\)
• \(\cot(90^\circ - \theta) = \tan \theta\)

In our exercise, these identities allow us to easily determine the remaining trigonometric functions using the known values of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\). Calculated as:

• \(\csc \theta = 1/\sin \theta\ \approx 1.4\)
• \(\sec \theta = 1/\cos \theta\ \approx 1.43\)
• \(\cot \theta = 1/\tan \theta\ \approx 0.98\)

By using these co-function relationships, you can derive functions effectively. They are particularly helpful when assessing various trigonometric functions quickly.

Acute Angles

An acute angle is any angle less than 90 degrees. In trigonometry, angles in the unit circle are typically expressed in radians, but for simplicity, degrees are often used in conventional explanations. For an acute angle \(\theta\), each trigonometric function value is positive:

• \(\sin \theta\)
• \(\cos \theta\)
• \(\tan \theta\)

These values represent different relationships between the sides of a right triangle. In our case, knowing \(\theta\) is acute gives us confidence that \(\sin \theta\) and \(\cos \theta\) must both be positive, which informs our solution:- Since \(\sin^2 \theta = 1 - \cos^2 \theta\), - \(\sin \theta\) is positive and equals \(0.714\).Understanding that the angle is acute helps eliminate the possibility of negative values, thus simplifying calculations. Remember:- Acute angles are foundational, helping to establish rules and identifications that guide trigonometric reasoning.