Question

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

Short answer

The remaining trigonometric functions are: \( \sin \theta = -\frac{3}{5}, \cos \theta = -\frac{4}{5}, \csc \theta = -\frac{5}{3}, \sec \theta = -\frac{5}{4}, \cot \theta = \frac{4}{3} \).

Step-by-step solution

Understand and Visualize the Given Information

The given information includes \(\tan \theta = \frac{3}{4}\) and \(\sin \theta < 0\). Note that \(\tan \theta\) is positive, which implies \(\theta\) is either in the 1st or 3rd quadrant. Since \(\sin \theta < 0\), \(\theta\) must be in the 3rd quadrant where sine is negative and tangent is positive.

Use the Definition of Tangent

Recall that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}\). Therefore, we can set \(y = 3\) and \(x = 4\), remembering that in the 3rd quadrant, both coordinates are negative. So, \(y = -3\) and \(x = -4\).

Determine the Hypotenuse

Use the Pythagorean Theorem to find \(r\), the hypotenuse: \(r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).

Calculate \( \sin \theta\)

Using the definition \(\sin \theta = \frac{y}{r}\), we get \(\sin \theta = \frac{-3}{5}\).

Calculate \( \cos \theta\)

Using the definition \( \cos \theta = \frac{x}{r} \), we get \( \cos \theta = \frac{-4}{5} \).

Calculate \( \csc \theta\)

Recall that \( \csc \theta = \frac{1}{\sin \theta} \), so \( \csc \theta = \frac{5}{-3} = -\frac{5}{3} \).

Calculate \( \sec \theta\)

Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so \( \sec \theta = \frac{5}{-4} = -\frac{5}{4} \).

Calculate \( \cot \theta\)

Using the definition \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{x}{y} \), we get \( \cot \theta = \frac{-4}{-3} = \frac{4}{3} \).

Key concepts

Tangent Function

The tangent function is a key concept in trigonometry, linking the opposite and adjacent sides of a right triangle through the ratio \(\tan \theta = \frac{y}{x} \). Here, \(y \) represents the length of the side opposite the angle \( \theta \) and \(x \) is the length of the side adjacent to the angle. Since we are given \( \tan \theta = \frac{3}{4} \), we can deduce the lengths of the opposite and adjacent sides based on this ratio. However, it's important to note the signs based on the quadrant: In this case, both should be negative because \( \theta \) lies in the 3rd quadrant, making \(y = -3 \) and \(x = -4 \).

Sine Function

The sine function relates the opposite side of a right triangle to the hypotenuse via the formula \( \sin \theta = \frac{y}{r} \). Given that \( \theta \) is in the 3rd quadrant where sine is negative and knowing \(y = -3\) along with the hypotenuse \(r \) calculated as 5, we get \( \sin \theta = \frac{-3}{5} \). The sine function is fundamental for understanding wave behavior, oscillations, and circular motions. It also complements the cosine function to provide a full picture of trigonometric relationships.

Cosine Function

Cosine, another primary trigonometric function, expresses the adjacent side of a right triangle with respect to the hypotenuse: \( \cos \theta = \frac{x}{r} \). From our example, with \(x = -4 \) and \(r = 5\), we calculate \( \cos \theta = \frac{-4}{5} \). The cosine function is essential in determining how far a point is along the x-axis, making it invaluable in various applications, including solving problems in physics and engineering that involve angles and distances.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry stating that for any right triangle, the square of the hypotenuse \((r)\) is equal to the sum of the squares of the other two sides: \[ r^2 = x^2 + y^2 \]. For our scenario, substituting \(x = -4\) and \(y = -3\), we find the hypotenuse as: \[ r = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]. This theorem not only aids in finding missing sides of triangles but also serves as a foundational tool across various areas of mathematics and science.