Question

Use identities to find the exact value of each of the four remaining trigonometric functions of the acute angle \(\theta\). $$\sin \theta=\frac{2}{3} \quad \cos \theta=\frac{\sqrt{5}}{3}$$

Short answer

The four remaining trigonometric functions are: \[ \tan \theta = \frac{2\sqrt{5}}{5}, \csc \theta = \frac{3}{2}, \sec \theta = \frac{3\sqrt{5}}{5}, \cot \theta = \frac{\sqrt{5}}{2} \]

Step-by-step solution

Verify Given Values using Pythagorean Identity

Use the Pythagorean identity to verify if the given values for \(\theta\) are correct. The identity states: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the given values: \[ \left( \frac{2}{3} \right)^2 + \left( \frac{\sqrt{5}}{3} \right)^2 = \frac{4}{9} + \frac{5}{9} = 1 \] The identity holds, so the given values are correct.

Find Tangent Function

The tangent of an angle is given by: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Substituting the given values: \[ \tan \theta = \frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \]

Find Cosecant Function

The cosecant of an angle is the reciprocal of the sine function: \[ \csc \theta = \frac{1}{\sin \theta} \] Substituting the given value: \[ \csc \theta = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

Find Secant Function

The secant of an angle is the reciprocal of the cosine function: \[ \sec \theta = \frac{1}{\cos \theta} \] Substituting the given value: \[ \sec \theta = \frac{1}{\frac{\sqrt{5}}{3}} = \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \]

Find Cotangent Function

The cotangent of an angle is the reciprocal of the tangent function: \[ \cot \theta = \frac{1}{\tan \theta} \] Substituting the value from Step 2: \[ \cot \theta = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2} \]

Key concepts

Pythagorean Identity

One of the fundamental trigonometric identities is the Pythagorean identity. This identity states that for any angle \theta\r, the sum of the square of the sine function and the square of the cosine function always equals 1. The formula is:\[ \sin^2 \theta + \cos^2 \theta = 1 \]Here's how it works for the given problem:

• Given the values \( \sin \theta = \frac {2}{3} \) and \( \cos \theta = \frac {\text {\s
• When we calculate \(( \frac{2}{3} )^2 + ( \frac{\sqrt{5}}{3} )^2 = \frac {4}{9} + \frac {5}{9} = 1\), the identity holds true, verifying that the given angles are correct.

Understanding and verifying the Pythagorean identity is crucial because it supports the integrity of the values given in trigonometric problems.

Tangent Function

The tangent function (tan) is a key trigonometric function that relates the sine and cosine of an angle. It's calculated by dividing sine by cosine:\[ \tan \theta = \frac{ \sin \theta } { \cos \theta } \]For the given values:

• We substitute \( \sin \theta = \frac {2}{3}\) and \(\cos\theta=\frac{\sqrt{5}}{3}\), leading to \( \tan \theta = \frac{ \frac {2}{3}} { \frac {\sqrt{5}}{3}} = \frac {2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}\).

This process shows how the tangent can be derived from sine and cosine, and it's useful for finding the slopes of angles in various applications.

Cosecant Function

The cosecant function (csc) is the reciprocal of the sine function. If you know the sine of an angle, the cosecant is simply:\[ \csc \theta = \frac {1} {\sin \theta }\]

• For \( \sin \theta = \frac{2}{3}\), we calculate \( \csc \theta = \frac{1}{\frac{2}{3}} = \frac{3}{2} \).

Understanding the cosecant function is essential, as it helps in various calculations where the sine function is involved, especially in problems dealing with right triangles and wave forms.

Secant Function

The secant function (sec) is the reciprocal of the cosine function and is defined by:\[ \sec \theta = \frac{1} {\cos \theta }\]

• Substituting \( \cos \theta = \frac{\sqrt{5}}{3} \) gives us \( \sec\theta = \frac{1}{\frac{\sqrt{5}}{3}} = \frac{3}{\sqrt{5}} = \frac{3 \sqrt{5}}{5} \).

The secant function is particularly useful in calculus, geometry, and in analyzing periodic functions such as waves.

Cotangent Function

The cotangent function (cot) is the reciprocal of the tangent function. It can also be defined as:\[ \cot \theta= \frac{1 } {\tan \theta } =\frac{\cos \theta } {\sin \theta } \]

• Given \( \tan \theta = \frac {2 \sqrt {5}}{ 5}\), we have \( \cot \theta = \frac{1}{ \frac {2\sqrt{5}}{5}} = \frac{5}{2 \sqrt{5}} = \frac{\sqrt{5}}{2} \).

The cotangent function is useful in solving triangles and modeling phenomena in engineering and physics.