Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the values of the six trigonometric functions of \(\theta\). Constraint \(\theta\) lies in Quadrant II. \(\theta\) lies in Quadrant III. \(\sin \theta < 0\) \(\cot \theta < 0\) \(0 \leq \theta \leq \pi\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\) \(\pi \leq \theta \leq 2 \pi\) Function Value $$\cos \theta=-\frac{4}{5}$$

Step-by-Step Solution

Answer

The six trigonometric function values are: cosθ = -4/5, sinθ = -3/5, tanθ = 3/4, cotθ = 4/3, secθ = -5/4, cscθ = -5/3.

Step 1: Calculate sinθ

From the Pythagorean identity: \( cos^2θ + sin^2θ = 1 \) \ , use the given value of cosθ (-4/5) to solve for sinθ. This gives us: \( sin^2θ = 1- cos^2θ = 1- (-4/5)^2 = 1- 16/25 = 9/25 \) . Because sin < 0 in the second quadrant, the value of sinθ is -3/5.

Step 2: Calculate tangent, cotangent, secant and cosecant

The formula for tangent (tanθ) is sinθ/cosθ, which gives: \( -3/5 / -4/5 = 3/4 \) . The formula for cotangent (cotθ) which equals to 1/tanθ is: \( 1 / 3/4 = 4/3 \) . The formula for secant (secθ) which equals to 1/cosθ is: \( 1 / -4/5 = -5/4 \) . The formula for cosecant which equals to 1/sinθ is: \( 1 / -3/5 = -5/3 \) .

Key Concepts

Quadrants in Trigonometry

Trigonometry often involves understanding the location of angles on the coordinate plane, called quadrants. The four quadrants are numbered I through IV and are arranged in a counter-clockwise direction starting from the positive x-axis. Each quadrant has distinct trigonometric sign characteristics:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, while cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, while sine and tangent are negative.

For the given problem, understanding these quadrant properties helps in determining signs for trigonometric values based on the constraints provided.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental in trigonometry. It states that for any angle \(\theta\), the square of the sine function plus the square of the cosine function equals one: \[\cos^2 \theta + \sin^2 \theta = 1\] This identity allows us to find a missing trigonometric ratio if one is known. For instance, with a given \(\cos \theta = -\frac{4}{5}\), you can find \(\sin \theta\) as: \[\sin^2 \theta = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25} \] When resolving for \(\sin \theta\) by taking the square root, remember to consider the quadrant-based sign of the result.

Trigonometric Identities

Trigonometric identities are equations that relate the six trigonometric functions with one another. Beyond the Pythagorean identity, the most basic include:

• Tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• Cotangent: \( \cot \theta = \frac{1}{\tan \theta} \)
• Secant: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cosecant: \( \csc \theta = \frac{1}{\sin \theta} \)

These identities make it convenient to derive unknown functions once one trigonometric value is known and play a crucial role in solving trigonometric equations. When working through problems, ensuring the accuracy of signs based on quadrant rules is essential.

Inverse Trigonometric Functions

Inverse trigonometric functions serve to find angles given a trigonometric function value. They are named arcus functions, like arcsin, arccos, and arctan. These functions return the arcs or angles from specific values and are useful when determining an angle's measure in stick rules, such as angles within specified interval constraints.