Problem 31 - Algebra and Trigonometry

Question

find the exact value of each of the remaining trigonometric functions of \(\theta\) $$ \tan \theta=\frac{4}{3}, \quad \cos \theta<0 $$

Step-by-Step Solution

Answer

The exact values for the trigonometric functions are: \(\sin \theta = \frac{4}{5}\), \(\cos \theta = -\frac{3}{5}\), \(\tan \theta = \frac{4}{3}\), \(\csc \theta = \frac{5}{4}\), \(\sec \theta = -\frac{5}{3}\), and \(\cot \theta = \frac{3}{4}\)

Step 1: Determine the quadrant of \(\theta\)

Considering that \(\tan \(\theta\) = \frac{4}{3}\) which is positive and \(\cos \(\theta\) is negative, the angle \(\theta\) must be in the second quadrant where the cosine is negative and tangent is positive.

Step 2: Construct a right triangle

In the second quadrant, create a right triangle where the tangent of \(\theta\) which is equal to \(\frac{opposite}{adjacent} = \frac{4}{3}\). The 'opposite' represents the y-coordinate and the 'adjacent' represents the x-coordinate in the second quadrant, hence we can consider this as (-3,4) which is based on the \(\tan \theta = \frac{4}{3}\) ratio and the cosine negative condition.

Step 3: Calculate the hypotenuse of the triangle

Use the Pythagorean theorem which says the square of the hypotenuse is equal to the sum of the squares of the other two sides. Hence, \(hypotenuse = \sqrt{(-3)^{2} + (4)^{2}} = 5\)

Step 4: Calculate the other trigonometric functions

With all sides of the triangle known, the trigonometric functions can be found as follows: \(\sin \theta = \frac{opposite}{hypotenuse} = \frac{4}{5}\), \(\cos \theta = \frac{adjacent}{hypotenuse} = -\frac{3}{5}\), \(\csc \theta = \frac{1}{\sin \theta} = \frac{5}{4}\), \(\sec \theta = \frac{1}{\cos \theta} = -\frac{5}{3}\) and \(\cot \theta = \frac{1}{\tan \theta} = \frac{3}{4}\)

Step 5: Verify the results

Double-check the trigonometric values computed. The negative values of cosine and secant and the positive values of sine, cosecant, tangent, and cotangent confirm that the angle \(\theta\) is indeed in the second quadrant.

Key Concepts

Quadrant Determination

Determining the quadrant is the first step to solving trigonometric problems like this.
A quadrant is a section of the Cartesian plane that is divided by the x and y axes.
There are four quadrants, and each has unique sign conventions for the trigonometric functions.

• The first quadrant: All trigonometric functions are positive.
• The second quadrant: Sine and cosecant are positive, while cosine, secant, tangent, and cotangent are negative.
• The third quadrant: Tangent and cotangent are positive, while sine, cosine, secant, and cosecant are negative.
• The fourth quadrant: Cosine and secant are positive, while sine, cosecant, tangent, and cotangent are negative.

In the problem, ewline \( \tan \theta = \frac{4}{3} \) is positive, and \( \cos \theta \) is negative, which places \( \theta \) in the second quadrant. Recognizing the signs helps determine the location in the graph for proper calculations.

Right Triangle Construction

Constructing a right triangle in the coordinate plane can help visualize and solve the problem.
A right triangle has one 90-degree angle, and in trigonometry, it helps in relating the sides of the triangle to angles using ratios.
For our problem, since \( \tan \theta = \frac{4}{3} \), the opposite side can be considered as 4, and the adjacent side as -3, reflecting the second quadrant conditions.
Right triangle construction in this scenario is crucial as it allows using these coefficients to find other trigonometric functions.

• Opposite side (y-coordinate): 4
• Adjacent side (x-coordinate): -3

Visualizing this correctly ensures proper application of trigonometric identities and theorems.

Pythagorean Theorem

The Pythagorean theorem is fundamental for calculating the hypotenuse of a right triangle.
The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides:
\[ c^2 = a^2 + b^2 \]
In our triangle, \( a = -3 \) and \( b = 4 \), so the hypotenuse \( c \) is calculated as:
\[ c = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5 \]
Knowing the hypotenuse is essential for finding any remaining trigonometric functions by relating it back to the sides of the triangle.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another.
In our triangle, having the sides and hypotenuse helps us find the remaining functions:

• \( \sin \theta = \frac{opposite}{hypotenuse} = \frac{4}{5} \)
• \( \cos \theta = \frac{adjacent}{hypotenuse} = -\frac{3}{5} \)
• \( \csc \theta = \frac{1}{\sin \theta} = \frac{5}{4} \)
• \( \sec \theta = \frac{1}{\cos \theta} = -\frac{5}{3} \)
• \( \cot \theta = \frac{1}{\tan \theta} = \frac{3}{4} \)

These identities confirm the quadrant decision and the consistent signs of each function. Understanding these relationships is key to accurately analyzing and solving trigonometric problems.