Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14,\) for each given function value, find the remaining five trigonometric function values. \(\csc \theta=\frac{5}{4}\) and \(\theta\) is in the second quadrant.

Step-by-Step Solution

Answer

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

Step 1: Understanding Reciprocal Functions

We start with the given function value, \(\csc \theta = \frac{5}{4}\). Remember that cosecant is the reciprocal of sine, so \(\sin \theta = \frac{1}{\csc \theta} = \frac{4}{5}\). Since \(\theta\) is in the second quadrant, where sine is positive, \(\sin \theta = \frac{4}{5}\).

Step 2: Using the Pythagorean Identity

Use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Plug in \(\sin \theta = \frac{4}{5}\): \[\left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1\]. \(\left(\frac{4}{5}\right)^2 = \frac{16}{25}\), so \(\cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}\).

Step 3: Solving for Cosine

Since \(\theta\) is in the second quadrant, cosine is negative here. Thus, \(\cos \theta = -\sqrt{\frac{9}{25}} = -\frac{3}{5}\).

Step 4: Finding the Other Trigonometric Functions

Now use sine and cosine to find the remaining trigonometric functions: - \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}\)- \(\cot \theta = \frac{1}{\tan \theta} = -\frac{3}{4}\)- \(\sec \theta = \frac{1}{\cos \theta} = -\frac{5}{3}\).

Key Concepts

Reciprocal Functions

Reciprocal functions in trigonometry are quite interesting and essential in understanding how different trigonometric functions relate to one another. In this exercise, \(\csc \theta\) is given as \(\frac{5}{4}\). The cosecant function, \(\csc \theta\), is the reciprocal of the sine function, \(\sin \theta\). This means that \(\sin \theta = \frac{1}{\csc \theta}\). By applying this relationship, we find \(\sin \theta = \frac{4}{5}\). This is crucial because knowing \(\sin \theta\) allows us to determine other trigonometric functions using their relationships.

• The reciprocal functions are: sine with cosecant (\(\sin \theta\) and \(\csc \theta\)), cosine with secant (\(\cos \theta\) and \(\sec \theta\)), and tangent with cotangent (\(\tan \theta\) and \(\cot \theta\)).
• These reciprocal relationships help simplify complex trigonometric expressions and solve equations.

Understanding reciprocal functions builds a foundation for solving various trigonometric problems, just like the one you've tackled here.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry that relates sine and cosine functions. In mathematical terms, the identity is expressed as \(\sin^2 \theta + \cos^2 \theta = 1\). This identity comes in handy for finding \(\cos \theta\) when you know \(\sin \theta\) and vice versa. When you know \(\sin \theta = \frac{4}{5}\), you can substitute it into the identity:\[\left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1\]Simplifying gives \(\frac{16}{25} + \cos^2 \theta = 1\). Solving for \(\cos^2 \theta\), we get \(\cos^2 \theta = \frac{9}{25}\).

• To find \(\cos \theta\), take the square root of both sides, yielding \(\cos \theta = \pm\sqrt{\frac{9}{25}}\).
• It’s crucial to consider the quadrant when determining the sign of trigonometric functions, as with \(\cos \theta\).

The Pythagorean identity helps relate angles to right triangles and is instrumental in trigonometric proofs and computations.

Second Quadrant

The concept of quadrants in the coordinate plane is vital in understanding the signs and values of trigonometric functions. Angles are measured from the positive x-axis, moving counterclockwise: first quadrant, second quadrant, third quadrant, and fourth quadrant. In this exercise, the angle \(\theta\) is positioned in the second quadrant. Each quadrant affects the sign of the trigonometric functions:

• In the second quadrant, sine (\(\sin \theta\)) is positive, but cosine (\(\cos \theta\)) and tangent (\(\tan \theta\)) are negative.
• This knowledge helps determine the correct signs for trigonometric functions when solving problems.

Understanding which quadrant an angle is in explains why \(\cos \theta = -\frac{3}{5}\) for this exercise. By knowing the positive and negative convention in each quadrant, you can accurately find and reason about any trigonometric function values.