Problem 42 - Algebra 2

Question

Suppose \(\theta\) is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of \(\theta .\) \(\cos \theta=\frac{3}{5},\) Quadrant IV

Step-by-Step Solution

Answer

\( \sin \theta = -\frac{4}{5}, \tan \theta = -\frac{4}{3}, \csc \theta = -\frac{5}{4}, \sec \theta = \frac{5}{3}, \cot \theta = -\frac{3}{4} \).

Step 1: Identify Cosine and Sign of Functions

Since \( \cos \theta = \frac{3}{5} \) and \( \theta \) is in the fourth quadrant, the cosine is positive. In the fourth quadrant, the sine function is negative, while the tangent, cotangent, and cosecant functions are also negative. Moreover, secant, being the reciprocal of cosine, is positive.

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that \( \sin^2 \theta + \cos^2 \theta = 1 \). Plug in the known value \( \cos \theta = \frac{3}{5} \) to find \( \sin \theta \):\[ \sin^2 \theta + \left( \frac{3}{5} \right)^2 = 1 \]\[ \sin^2 \theta + \frac{9}{25} = 1 \]\[ \sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \]Take the negative square root since sine is negative in the fourth quadrant: \[ \sin \theta = -\frac{4}{5} \]

Step 3: Find Remaining Trigonometric Functions

Using the known values of \( \sin \theta \) and \( \cos \theta \):- \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3} \)- \( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \)- \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \)- \( \cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{4}{3}} = -\frac{3}{4} \)

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental relationships in trigonometry. It expresses the connection between the sine and cosine functions. The identity is given by: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This means that for any angle \( \theta \), the square of the sine of \( \theta \) added to the square of the cosine of \( \theta \) will always equal one.
This identity is derived from the famous Pythagorean theorem applied within the unit circle. In the unit circle, any point can be represented by \( (\cos \theta, \sin \theta) \). Since the radius of the unit circle is 1, using the Pythagorean theorem gives \( (\cos \theta)^2 + (\sin \theta)^2 = 1^2 \).
This identity is crucial for solving trigonometric equations and understanding the relationship between the functions of angles.

cosine function

The cosine function, often denoted as \( \cos \theta \), is a primary trigonometric function. It represents the x-coordinate of a point on the unit circle. The value of \( \cos \theta \) varies as \( \theta \) changes, ranging from -1 to 1.

• When \( \theta \) is 0 degrees, \( \cos \theta \) is 1.
• When \( \theta \) is 90 degrees, \( \cos \theta \) is 0.
• When \( \theta \) is 180 degrees, \( \cos \theta \) is -1.
• When \( \theta \) is 270 degrees, \( \cos \theta \) turns back to 0.

The cosine function is especially useful for understanding the behavior of points on the circle and in solving different practical problems in physics and engineering.
It also possesses even symmetry, which means \( \cos(-\theta) = \cos(\theta) \), making it unique among trigonometric functions.

sine function

The sine function, denoted as \( \sin \theta \), is another primary trigonometric function that represents the y-coordinate of a point on the unit circle. Like cosine, the sine function has values that range from -1 to 1 as the angle \( \theta \) varies.

• At \( \theta = 0^\circ \), \( \sin \theta = 0 \).
• At \( \theta = 90^\circ \), \( \sin \theta = 1 \).
• At \( \theta = 180^\circ \), \( \sin \theta = 0 \).
• At \( \theta = 270^\circ \), \( \sin \theta = -1 \).

The sine function is crucial for modeling periodic phenomena such as sound waves, light waves, and tides.
Unlike cosine, the sine function has odd symmetry, meaning \( \sin(-\theta) = -\sin(\theta) \), which can be particularly useful in solving equations and understanding wave behavior.

fourth quadrant

In the Cartesian coordinate system, the fourth quadrant is found in the lower right corner. Here, the x-values are positive, and the y-values are negative. This positioning affects the signs of trigonometric functions.

• The cosine function is positive in the fourth quadrant, which explains why \( \cos \theta = \frac{3}{5} \) is positive.
• The sine function is negative, so \( \sin \theta = -\frac{4}{5} \).
• Tangent, cosecant, and cotangent are all negative in this quadrant.
• Secant remains positive since it's the reciprocal of the positive cosine.

Understanding the signs of trigonometric functions based on quadrant is essential in trigonometry as it helps determine the exact values of various trigonometric functions for angles that don't lie on the x- or y-axis.