Problem 31 - Precalculus Essentials

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

\(\sin \theta = -4/5\), \(\cos \theta = -3/5\), \(\csc \theta = -5/4\), \(\sec \theta = -5/3\), \(\cot \theta = 3/4\)

Step 1: Determine Sine and Cosine

For any right triangle with sides a (adjacent), b (opposite), and c (hypotenuse), we have \(\tan \theta = \frac{b}{a}\). Here, given \(\tan \theta = \frac{4}{3}\), we can assume a=3 and b=4 (identifying a as adjacent and b as opposite, considering the third quadrant). We can then use the Pythagorean theorem (a² + b² = c²) to solve for c, i.e., \(c = \sqrt{(3)^2 + (4)^2} = 5\). Now we could find \(\sin \theta=\frac{b}{c}\) = -4/5 and \(\cos \theta =\frac{a}{c}\) = -3/5. Remember, both sine and cosine are negative in the third quadrant.

Step 2: Determine Remaining Trigonometric Functions

Having found sine and cosine, we can now easily calculate other trigonometric functions. It's important to remember their formulas: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\) or \(\cot \theta =\frac{\cos \theta}{\sin \theta}\). Substitute the respective values to get \(\csc \theta = \frac{1}{-4/5} = -5/4\), \(\sec \theta = \frac{1}{-3/5} = -5/3\), and \(\cot \theta = \frac{1}{4/3} = 3/4\).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry related to right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula:

• \[ c^2 = a^2 + b^2 \]

In this exercise, we use the Pythagorean theorem to find the hypotenuse when given the two legs of a right triangle. Assuming one leg to be 3 and the other to be 4, we plug these values into the formula to get the hypotenuse:

• \[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \]

Understanding this theorem helps us solve many trigonometric problems, especially when we deal with angles and side lengths.

third quadrant

The third quadrant of the coordinate system is where both the x and y values are negative. In the context of trigonometric functions, this means both the sine and cosine values are negative.

• The angle \( \theta \) is assumed to be in this quadrant based on the given conditions: \( \tan \theta = \frac{4}{3} \) (positive) and \( \cos \theta < 0 \).
• As tangent is positive and cosine is negative, sine must also be negative, which confirms the location in the third quadrant where both sine and cosine have negative values.

Knowing the quadrant of an angle is crucial because it directly affects the signs of trigonometric functions, helping determine the proper values of these functions.

tangent function

The tangent function in trigonometry is defined as the ratio of the opposite side to the adjacent side in a right triangle. It is expressed as:

• \[ \tan \theta = \frac{b}{a} \]

In the exercise, we are given \( \tan \theta = \frac{4}{3} \). This reveals the relative lengths of the opposite and adjacent sides when \( \theta \) is positioned in a right triangle in the third quadrant.

• Because \( \tan \theta \) is positive and \( \cos \theta < 0 \), it indicates that both sine and cosine are negative in the third quadrant, supporting their negative values.

Understanding tangent helps in calculating the other trigonometric functions through relationships and identities.

sine and cosine

In trigonometry, the sine and cosine functions are fundamental. For an angle \( \theta \) in a right triangle:

• Sine is the ratio of the length of the opposite side to the hypotenuse, expressed as \( \sin \theta = \frac{b}{c} \).
• Cosine is the ratio of the length of the adjacent side to the hypotenuse, expressed as \( \cos \theta = \frac{a}{c} \).

In the problem, after using the Pythagorean theorem, the hypotenuse is found to be 5. Thus, the sine and cosine of \( \theta \) are:

• \( \sin \theta = -\frac{4}{5} \)
• \( \cos \theta = -\frac{3}{5} \)

Both values are negative, consistent with being in the third quadrant, where sine and cosine always take on negative values. These functions form the basis for other trigonometric calculations.

reciprocal identities

Reciprocal identities are relationships between trigonometric functions that help solve problems more efficiently. They connect primary functions with their reciprocal functions:

• Cosecant is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent is the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \) or \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

In the exercise, we calculate:

• \( \csc \theta = -\frac{5}{4} \)
• \( \sec \theta = -\frac{5}{3} \)
• \( \cot \theta = \frac{3}{4} \)

These formulas are especially useful when verifying or deriving trigonometric expressions and solving complex problems.