Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of \(\sin 2 \theta, \cos 2 \theta\), and \(\tan 2 \theta\) for the given values of \(\theta\). $$ \sec \theta=-3 ; \quad 90^{\circ}<\theta<180^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{4\sqrt{2}}{9}\), \(\cos 2\theta = -\frac{7}{9}\), \(\tan 2\theta = \frac{4\sqrt{2}}{7}\).

Step 1: Determine \\(\cos \theta\\)

The reciprocal of \( \sec \theta \) is \( \cos \theta \). Since \( \sec \theta = -3 \), this means \( \cos \theta = -\frac{1}{3} \).

Step 2: Find \\(\sin \theta\\) using Pythagorean Identity

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = -\frac{1}{3} \) into the equation:\[ \sin^2 \theta + \left(-\frac{1}{3}\right)^2 = 1 \]Simplify to find \( \sin \theta \):\[ \sin^2 \theta + \frac{1}{9} = 1 \]\[ \sin^2 \theta = \frac{8}{9} \]\[ \sin \theta = \pm \frac{\sqrt{8}}{3} = \pm \frac{2\sqrt{2}}{3} \]Since \( 90^\circ < \theta < 180^\circ \) places \( \theta \) in the second quadrant, where \( \sin \theta \) is positive, choose \( \sin \theta = \frac{2\sqrt{2}}{3} \).

Step 3: Calculate \\(\sin 2 \theta\\)

Use the double angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \).Substitute \( \sin \theta = \frac{2\sqrt{2}}{3} \) and \( \cos \theta = -\frac{1}{3} \):\[ \sin 2\theta = 2 \left( \frac{2\sqrt{2}}{3} \right) \left( -\frac{1}{3} \right) \]\[ \sin 2\theta = -\frac{4\sqrt{2}}{9} \]

Step 4: Calculate \\(\cos 2 \theta\\)

Use the double angle identity for cosine: \( \cos 2\theta = 2 \cos^2 \theta - 1 \).Substitute \( \cos \theta = -\frac{1}{3} \):\[ \cos 2\theta = 2 \left(-\frac{1}{3}\right)^2 - 1 \]\[ \cos 2\theta = 2 \left(\frac{1}{9}\right) - 1 \]\[ \cos 2\theta = \frac{2}{9} - 1 \]\[ \cos 2\theta = -\frac{7}{9} \]

Step 5: Calculate \\(\tan 2 \theta\\)

Use the double angle identity for tangent: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).First, calculate \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = -2\sqrt{2} \).Then substitute into the double angle formula:\[ \tan 2\theta = \frac{2(-2\sqrt{2})}{1 - (-2\sqrt{2})^2} \]\[ \tan 2\theta = \frac{-4\sqrt{2}}{1 - 8} \]\[ \tan 2\theta = \frac{-4\sqrt{2}}{-7} \]\[ \tan 2\theta = \frac{4\sqrt{2}}{7} \]

Key Concepts

Double Angle Identities

Understanding double angle identities is important for solving trigonometric problems efficiently. These identities express trigonometric functions of double angles, such as \(2\theta\), in terms of single angles, \(\theta\). This allows us to calculate expressions like \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\) using known values of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).

• The double angle identity for sine is: \( \sin 2\theta = 2 \sin \theta \cos \theta \).
• For cosine, we have: \( \cos 2\theta = 2 \cos^2 \theta - 1 \).
• The tangent double angle identity is: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).

By applying these identities, we can transform and simplify complex trigonometric expressions, making them easier to work with.

Pythagorean Identity

The Pythagorean identity is a fundamental concept that relates the square of sine and cosine to 1. It is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \), forming the basis for many trigonometric computations.
This identity is particularly useful when one of the sine or cosine values is known, and you need to determine the other. For instance, if you know \( \cos \theta = -\frac{1}{3} \), you can use the identity to find \( \sin \theta \).

• Substitute known values into the identity to solve for the unknown: \( \sin^2 \theta + \left( -\frac{1}{3} \right)^2 = 1 \).
• Simplify to get \( \sin^2 \theta = \frac{8}{9} \), and since the angle \(\theta\) is in the second quadrant, where sine is positive, take \( \sin \theta = \frac{2\sqrt{2}}{3} \).

This process shows how the Pythagorean identity helps to derive the exact trigonometric function values required for further calculation.

Sine Function

The sine function measures the y-coordinate or vertical component of a point on the unit circle corresponding to an angle \(\theta\). It is one of the primary trigonometric functions and has various applications in geometry and physics.
To find the sine value, especially when given only partial information like \(\sec \theta\), we often rely on the Pythagorean identity or double angle identities.

• In the problem, since \(90^{\circ} < \theta < 180^{\circ}\), \(\theta\) lies in the second quadrant where sine values are positive.
• After calculating using \( \sin^2 \theta = \frac{8}{9} \), it confirms \( \sin \theta = \frac{2\sqrt{2}}{3} \).

Recognizing the appropriate quadrant is crucial in determining when sine should be negative or positive.

Cosine Function

The cosine function represents the x-coordinate or horizontal component of a point on the unit circle corresponding to an angle \(\theta\). It is equally crucial alongside sine in solving trigonometry problems.
Given that \(\sec \theta = -3\), we find \(\cos \theta = -\frac{1}{3}\). This value of cosine was determined as \(\theta\) is in the second quadrant where cosine values are typically negative.

• The cosine double angle identity, \( \cos 2\theta = 2 \cos^2 \theta - 1 \), was used to find \( \cos 2\theta = -\frac{7}{9} \).
• Calculating this involves substituting the value \( \cos \theta = -\frac{1}{3} \).

Understanding how cosine behaves with respect to angle positions and double angle manipulation allows us to perform accurate transformations.

Tangent Function

The tangent function expresses the ratio of the sine function to the cosine function, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The tangent is especially useful for calculating angles and solving geometric problems.
When determining \(\tan 2\theta\), we use the double angle identity: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).

• First, find \( \tan \theta \) using the ratio \( \tan \theta = \frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = -2\sqrt{2} \).
• Then apply \( \tan 2\theta = \frac{-4\sqrt{2}}{-7} \), resulting in \( \tan 2\theta = \frac{4\sqrt{2}}{7} \).

Comprehending the interplay between sine, cosine, and tangent through the double angle identity permits precise computation of tangent values for different scenarios.