Problem 20 - Algebra and Trigonometry

Question

\(\theta\) is an acute angle and sin u and cos u are given. Use identities to find tan \(\theta\), csc \(\theta\), sec \(\theta\), and cot \(\theta\). Where necessary, rationalize denominators. $$ \sin \theta=\frac{6}{7}, \quad \cos \theta=\frac{\sqrt{13}}{7} $$

Step-by-Step Solution

Answer

The values are: tan θ = \(\frac{6\sqrt{13}}{13}\), csc θ = \(\frac{7}{6}\), sec θ = \(\frac{7\sqrt{13}}{13}\), and cot θ = \(\frac{\sqrt{13}}{6}\).

Step 1: Compute Tan θ

To find tan θ, recall the Pythagorean identity that states tan θ = \(\frac{\sin \theta}{\cos \theta}\). Now, substitute the given values into the equation: tan θ = \(\frac{6/7}{\sqrt{13}/7}\) = \(\frac{6}{\sqrt{13}}\). Rationalize the denominator to have tan θ = \(\frac{6\sqrt{13}}{13}\).

Step 2: Compute Csc θ

To find csc θ, use the reciprocal identity that states csc θ = \(\frac{1}{\sin \theta} \). Substitute the given value into the equation: csc θ = \(\frac{1}{6/7}\) = \(\frac{7}{6}\).

Step 3: Compute Sec θ

To find sec θ, apply the reciprocal identity, sec θ = \(\frac{1}{\cos \theta}\). Substitute the provided value into the equation: sec θ = \(\frac{1}{\sqrt{13}/7}\) = \(\frac{7}{\sqrt{13}}\). After rationalizing the denominator, you get sec θ = \(\frac{7\sqrt{13}}{13}\).

Step 4: Compute Cot θ

To find cot θ, use the reciprocal of tan θ equation: cot θ = \(\frac{1}{\tan \theta} \). Substitute the previously calculated value into the formula: cot θ = \(\frac{1}{6/\sqrt{13}}\) = \(\frac{\sqrt{13}}{6}\).

Key Concepts

Acute Angles

An acute angle is one that measures less than 90 degrees. In trigonometry, working with acute angles involves understanding the specific behaviors of sine, cosine, and tangent functions within this range. These angles typically appear in right-angle triangles, where they are the non-right angles.

For example, when dealing with angles like \( \theta \) in the exercise, which is stated to be acute, this means:

• \( \theta \) is less than 90 degrees.
• Both sine and cosine values are positive, as the angle is in the first quadrant of the unit circle.

Understanding acute angles is crucial because they form the basis of many trigonometric calculations and identities. As angles increase beyond 90 degrees, their sine, cosine, and tangent values begin to behave differently, demonstrating negative or opposite magnitude characteristics.

Rationalizing Denominators

Rationalizing the denominator is a process used in mathematics to eliminate radicals (like square roots) from the bottom of a fraction. It makes expressions easier to read and work with.

Here's how you do it: if you have a fraction \( \frac{a}{\sqrt{b}} \), you multiply both the numerator and denominator by \( \sqrt{b} \) to clear the radical from the denominator:

• Start with \( \frac{6}{\sqrt{13}} \).
• Multiply both numerator and denominator by \( \sqrt{13} \):
• This gives you \( \frac{6\sqrt{13}}{13} \).

Rationalizing helps to simplify expressions for processes like addition, subtraction, or division between different fractions. Furthermore, it provides clearer insights when interpreting results, as fractions with rationalized denominators are considered standard form.

Reciprocal Identities

Reciprocal identities are a set of simple trigonometric identities that relate a function to its reciprocal. These are helpful in calculating trigonometric functions indirectly. Here are the basic reciprocal identities:

• Cosecant \( \theta \) is the reciprocal of sine \( \theta \): \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant \( \theta \) is the reciprocal of cosine \( \theta \): \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent \( \theta \) is the reciprocal of tangent \( \theta \): \( \cot \theta = \frac{1}{\tan \theta} \).

These identities allow for quick conversion between functions, which is advantageous when solving trigonometric equations across various contexts.

When used correctly, they simplify complex fractions and make it feasible to switch from sine, cosine, or tangent to their respective reciprocal functions seamlessly. This proves especially useful, as seen in the task of finding trigonometric values from given sine or cosine measurements, incorporating rationalization for clarity in the final expressions.