Problem 53 - Basic Technical Mathematics with Calculus

Question

Find the values of the indicated trigonometric functions. Find \(\tan \theta,\) given \(\sec \theta=1.3698\)

Step-by-Step Solution

Answer

\( \tan \theta \approx 0.936 \)

Step 1: Understand the Relationship between Secant and Cosine

The secant function is the reciprocal of the cosine function. Therefore, if \( \sec \theta = 1.3698 \), then \( \cos \theta = \frac{1}{1.3698} \). Calculate this value to begin finding the tangent function.

Step 2: Calculate the Value of Cosine

Calculate \( \cos \theta \) using the reciprocal relationship: \( \cos \theta = \frac{1}{1.3698} \approx 0.730 \). This represents the cosine of the angle \( \theta \).

Step 3: Use the Pythagorean Identity to Find Sine

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), solve for \( \sin \theta \). Substitute \( \cos \theta = 0.730 \) to find \( \sin \theta \): \( \sin^2 \theta = 1 - (0.730)^2 \).

Step 4: Calculate Value of Sine

Compute \( \sin^2 \theta = 1 - 0.5329 = 0.4671 \), thus \( \sin \theta = \sqrt{0.4671} \approx 0.6836 \).

Step 5: Calculate the Value of Tangent

Now that we have both sine and cosine, use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute \( \sin \theta = 0.6836 \) and \( \cos \theta = 0.730 \) to find \( \tan \theta \).

Step 6: Final Calculation of Tangent

Calculate \( \tan \theta = \frac{0.6836}{0.730} \approx 0.936 \). This is the value of \( \tan \theta \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry that connects the sine and cosine of an angle. It is crucial for solving many trigonometric problems. Written as \[ \sin^2 \theta + \cos^2 \theta = 1 \]this equation allows you to find one trigonometric function when the other is known. Consider this identity as a tool for checking calculations and determining values.
For instance, if you already have \(\cos \theta\) (like we do in our exercise with \(\cos \theta \approx 0.730\)), you can rearrange the identity to find \(\sin \theta\). Plugging in the value, you get \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - 0.5329 = 0.4671 \]which gives \[ \sin \theta = \sqrt{0.4671} \approx 0.6836 \]Add this to your toolbox for solving various trigonometric equations efficiently.

Secant and Cosine Relationship

Understanding the connection between the secant and cosine functions can simplify trigonometric problems. The secant function \(\sec \theta\) is simply the reciprocal of the cosine function \(\cos \theta\). Thus, if given \(\sec \theta\), you can find \(\cos \theta\) using the formula: \[ \cos \theta = \frac{1}{\sec \theta} \]In our exercise, since \(\sec \theta = 1.3698\), you would calculate \[ \cos \theta = \frac{1}{1.3698} \approx 0.730 \]This step is essential, as having the value of the cosine is often a prerequisite for utilizing identities or computing other trigonometric functions. Remember, knowing that one function leads directly to another can save time and effort.

Tangent Calculation

After determining both the sine and cosine, calculating the tangent becomes straightforward. The tangent function is defined as the ratio of the sine to the cosine: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]In the context of our problem, where \(\sin \theta \approx 0.6836\) and \(\cos \theta \approx 0.730\), you calculate \[ \tan \theta = \frac{0.6836}{0.730} \approx 0.936 \]The tangent tells us how steeply a line rises or falls when it deviates from the horizontal axis.
Comprehending tangent alongside sine and cosine is vital because it aids in solving right triangle problems, determining angles, and analyzing waves. Always verify your tangent calculations against the sine and cosine values to ensure they align with the expected trigonometric identities.