Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14,\) for each given function value, find the remaining five trigonometric function values. \(\cos \theta=\frac{3}{4}\) and \(\theta\) is in the first quadrant.

Step-by-Step Solution

Answer

\(\sin \theta = \frac{\sqrt{7}}{4}, \tan \theta = \frac{\sqrt{7}}{3}, \cot \theta = \frac{3\sqrt{7}}{7}, \sec \theta = \frac{4}{3}, \csc \theta = \frac{4\sqrt{7}}{7}\).

Step 1: Find sine value

Since \(\theta\) is in the first quadrant, all trigonometric values are positive. We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). We know \(\cos \theta = \frac{3}{4}\). Thus, \(\sin^2 \theta + \left(\frac{3}{4}\right)^2 = 1\). Therefore, \(\sin^2 \theta = 1 - \frac{9}{16}\), giving \(\sin^2 \theta = \frac{7}{16}\). Taking the positive square root (since \(\theta\) is in the first quadrant), \(\sin \theta = \frac{\sqrt{7}}{4}\).

Step 2: Find tangent value

Tangent is the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the known values, \(\tan \theta = \frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}} = \frac{\sqrt{7}}{3}\).

Step 3: Find cotangent value

Cotangent is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta}\). Therefore, \(\cot \theta = \frac{3}{\sqrt{7}}\). To rationalize the denominator, multiply numerator and denominator by \(\sqrt{7}\) to get \(\cot \theta = \frac{3\sqrt{7}}{7}\).

Step 4: Find secant value

Secant is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta}\). Thus, \(\sec \theta = \frac{4}{3}\).

Step 5: Find cosecant value

Cosecant is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\csc \theta = \frac{4}{\sqrt{7}}\). To rationalize the denominator, multiply numerator and denominator by \(\sqrt{7}\), getting \(\csc \theta = \frac{4\sqrt{7}}{7}\).

Key Concepts

Pythagorean Identity

One of the cornerstone identities in trigonometry is the Pythagorean identity. It is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity comes from the famous Pythagorean theorem applied to a unit circle, where the hypotenuse (radius of the circle) is 1.
In a unit circle, any point on the circle has coordinates \(( \cos \theta, \sin \theta )\). The sum of the squares of these coordinates equals 1 because they form the two sides of a right triangle with the unit radius as the hypotenuse.

Thus, knowing either sine or cosine allows you to find the other using this identity. In our example with \( \cos \theta = \frac{3}{4} \), we used this identity to find \( \sin \theta = \frac{\sqrt{7}}{4} \). Just plug the known value into the identity and solve for the other trigonometric function.

Trigonometric Functions

These are functions of an angle used to relate angles in a right triangle to the ratios of the sides. There are six main trigonometric functions:

• Sine (sin): ratio of opposite side to hypotenuse.
• Cosine (cos): ratio of adjacent side to hypotenuse.
• Tangent (tan): ratio of opposite side to adjacent side.
• Cotangent (cot): reciprocal of tangent.
• Secant (sec): reciprocal of cosine.
• Cosecant (csc): reciprocal of sine.

These functions are primarily defined using the geometry of a right triangle or the unit circle.
For example, in the original exercise, by knowing \( \cos \theta \), we used these function definitions to determine \( \sin \theta \), \( \tan \theta \), and the reciprocal functions.

They help us connect the angle itself to the sides of a triangle or the coordinates of a unit circle point, which can model many real-world cyclic phenomena.

Reciprocal Identities

Reciprocal identities allow us to express some trigonometric functions in terms of others. They are particularly useful when certain function values are given, and you wish to find others.

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

These identities transform more complex fractions into simpler forms or vice-versa, and facilitate calculations.
In our problem, after finding \( \sin \theta \) and \( \cos \theta \), we used these reciprocal identities to derive \( \sec \theta \), \( \csc \theta \), and \( \cot \theta \).

Rationalizing the denominator is a common step when using these identities, as seen in the calculations of \( \csc \theta \) and \( \cot \theta \). Understanding how these identities work will make solving trigonometry problems much easier.