Problem 85 - Algebra and Trigonometry with Analytic Geometry

Question

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\tan \theta=-\frac{3}{4}\( and \)\sin \theta>0$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Conditions

We are given that \( \tan \theta = -\frac{3}{4} \) and \( \sin \theta > 0 \). This implies that \( \theta \) is in the second quadrant because, in the second quadrant, tangent is negative and sine is positive.

Step 2: Use the Pythagorean Identity for Tangent

Recall the identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, \( \sin \theta = -\frac{3}{4} \cos \theta \).

Step 3: Use the Pythagorean Identity for Sine and Cosine

The Pythagorean identity states \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = -\frac{3}{4} \cos \theta \) into the identity: \[ \left( -\frac{3}{4} \cos \theta \right)^2 + \cos^2 \theta = 1 \].

Step 4: Simplify and Solve the Equation

Simplify \( \left( -\frac{3}{4} \cos \theta \right)^2 = \frac{9}{16} \cos^2 \theta \), then the equation is \[ \frac{9}{16} \cos^2 \theta + \cos^2 \theta = 1 \]. Factor \( \cos^2 \theta \): \[ \left( \frac{9}{16} + 1 \right)\cos^2 \theta = 1 \].

Step 5: Final Calculation for \( \cos \theta \)

Combine terms: \( \frac{9}{16} + \frac{16}{16} = \frac{25}{16} \). So, the equation becomes \( \frac{25}{16} \cos^2 \theta = 1 \). Solve for \( \cos^2 \theta \): \[ \cos^2 \theta = \frac{16}{25} \].

Step 6: Determine \( \cos \theta \) Using the Quadrant Information

Since \( \cos^2 \theta = \frac{16}{25} \), we have two possibilities: \( \cos \theta = \pm\frac{4}{5} \). In the second quadrant, \( \cos \theta \) is negative. Therefore, \( \cos \theta = -\frac{4}{5} \).

Step 7: Determine \( \sin \theta \)

Use \( \sin^2 \theta + \cos^2 \theta = 1 \) with \( \cos^2 \theta = \frac{16}{25} \): \( \sin^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} \). Thus \( \sin \theta = \pm\frac{3}{5} \). Since we are in the second quadrant, \( \sin \theta > 0 \), so \( \sin \theta = \frac{3}{5} \).

Step 8: Verify and List All Trigonometric Functions

We have \( \sin \theta = \frac{3}{5} \), \( \cos \theta = -\frac{4}{5} \), and \( \tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{3}{4} \). The other functions can be found: \( \csc \theta = \frac{1}{\sin \theta} = \frac{5}{3} \), \( \sec \theta = \frac{1}{\cos \theta} = -\frac{5}{4} \), and \( \cot \theta = \frac{1}{\tan \theta} = -\frac{4}{3} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), the square of the sine function plus the square of the cosine function will always equal one: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is derived from the Pythagorean Theorem, which applies to right-angled triangles. By considering a unit circle (a circle with radius 1), the coordinates \((x, y)\) on the circle can be represented by \((\cos \theta, \sin \theta)\), confirming the identity.
This identity is powerful for solving various trigonometric problems because:

• You can determine one trigonometric function by knowing the other.
• It's universally applicable to all angles \(\theta\).

By rearranging and using this identity, other relationships such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) can be explored. It's often used to verify and solve equations in trigonometric contexts.

Trigonometric Functions

Trigonometric functions are essential in understanding relationships between angles and sides in triangles. The primary trigonometric functions are Sine (\(\sin\)), Cosine (\(\cos\)), and Tangent (\(\tan\)). Each function has an associated reciprocal: Cosecant (\(\csc\)), Secant (\(\sec\)), and Cotangent (\(\cot\)).
These functions are defined as follows:

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

Reciprocal functions flip the fraction:

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

The tangent is particularly interesting because it can also be expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Knowing these fundamental definitions allows you to link angles and side lengths in trigonometric problems, ensuring mastery over basic and more advanced principles.

Quadrants in Trigonometry

The unit circle is divided into four quadrants, which are important for understanding the behavior of trigonometric functions. Each quadrant has distinct characteristics that affect the signs of the trigonometric functions, crucial for solving trigonometric problems.
The quadrants are numbered counterclockwise starting from the positive x-axis:

• **Quadrant I:** All trigonometric functions are positive.
• **Quadrant II:** \(\sin\) is positive while \(\cos\) and \(\tan\) are negative.
• **Quadrant III:** \(\tan\) is positive while \(\sin\) and \(\cos\) are negative.
• **Quadrant IV:** \(\cos\) is positive while \(\sin\) and \(\tan\) are negative.

Understanding in which quadrant an angle lies can help with determining the correct sign for the function values. For instance, knowing \(\tan \theta = -\frac{3}{4} \) and \(\sin \theta > 0 \) places \(\theta\) in the second quadrant, where these conditions naturally occur. This knowledge aids in ensuring equations are solved correctly and logically.