Problem 20 - Algebra 2 and Trigonometry

Question

a. Find the exact value of \(\cos 210^{\circ}\) by using \(\cos \left(270^{\circ}-60^{\circ}\right) .\) b. Find the exact value of \(\sin 210^{\circ}\) by using \(\cos ^{2} \theta+\sin ^{2} \theta=1\) and the value of \(\cos 210^{\circ}\) found in a. c. Find the exact value of \(\cos 165^{\circ}\) by using \(\cos \left(210^{\circ}-45^{\circ}\right) .\) d. Use the value of \(\cos 165^{\circ}\) found in \(c\) to find \(\cos \left(-15^{\circ}\right)\) by using \(\cos \left(165^{\circ}-180^{\circ}\right)\) e. Use the value of \(\cos \left(-15^{\circ}\right)\) found in \(\mathbf{d}\) to find \(\cos 195^{\circ}\) by using \(\cos \left(180^{\circ}-\left(-15^{\circ}\right)\right) .\) f. Use the value of \(\cos \left(-15^{\circ}\right)\) found in \(\mathbf{d}\) to find the exact value of \(\sin 105^{\circ} .\)

Step-by-Step Solution

Answer

\( \cos 210^{\circ} = -\frac{\sqrt{3}}{2} \), \( \sin 210^{\circ} = -\frac{1}{2} \), \( \cos 165^{\circ} = -\frac{\sqrt{6} + \sqrt{2}}{4} \), \( \cos -15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \), \( \sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \)

Step 1: Use Cosine Angle Difference Formula

To find \( \cos(210^{\circ}) \) using \( \cos(270^{\circ} - 60^{\circ}) \), apply the cosine difference identity: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]Substitute \( a = 270^{\circ} \) and \( b = 60^{\circ} \):\[ \cos(210^{\circ}) = \cos(270^{\circ})\cos(60^{\circ}) + \sin(270^{\circ})\sin(60^{\circ}) \]Since \( \cos(270^{\circ}) = 0 \) and \( \sin(270^{\circ}) = -1 \):\[ \cos(210^{\circ}) = 0 \cdot \frac{1}{2} + (-1) \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} \]

Step 2: Use Pythagorean Identity to Find Sine

Now, find \( \sin(210^{\circ}) \) using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):\[ \sin^2(210^{\circ}) = 1 - \cos^2(210^{\circ}) \]Substitute \( \cos(210^{\circ}) = -\frac{\sqrt{3}}{2} \):\[ \sin^2(210^{\circ}) = 1 - \left(-\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4} \]Thus, \( \sin(210^{\circ}) = -\frac{1}{2} \) (since \(210^{\circ}\) is in the third quadrant where sine is negative).

Step 3: Use Cosine Angle Difference for 165 degrees

To find \( \cos(165^{\circ}) \) using \( \cos(210^{\circ} - 45^{\circ}) \), use the cosine difference identity again:\[ \cos(165^{\circ}) = \cos(210^{\circ})\cos(45^{\circ}) + \sin(210^{\circ})\sin(45^{\circ}) \]Substitute values:\( \cos(210^{\circ}) = -\frac{\sqrt{3}}{2} \), \( \sin(210^{\circ}) = -\frac{1}{2} \), \( \cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \):\[ \cos(165^{\circ}) = -\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} + -\frac{1}{2}\cdot\frac{\sqrt{2}}{2} = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4} \]

Step 4: Use Cosine Angle Difference for -15 degrees

To find \( \cos(-15^{\circ}) \) using \( \cos(165^{\circ} - 180^{\circ}) \), use the formula again:\[ \cos(-15^{\circ}) = \cos(165^{\circ})\cos(180^{\circ}) + \sin(165^{\circ})\sin(180^{\circ}) \]Since \( \cos(180^{\circ}) = -1 \) and \( \sin(180^{\circ}) = 0 \):\[ \cos(-15^{\circ}) = \cos(165^{\circ})\cdot(-1) + 0\cdot\sin(165^{\circ}) = -(-\frac{\sqrt{6} + \sqrt{2}}{4}) = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Step 5: Find Cosine of 195 degrees

Now, use \( \cos(180^{\circ} - (-15^{\circ})) = \cos(195^{\circ}) \):\[ \cos(195^{\circ}) = -\cos(-15^{\circ}) \]Substitute \( \cos(-15^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4} \):\[ \cos(195^{\circ}) = -\frac{\sqrt{6} + \sqrt{2}}{4} \]

Step 6: Use Reference Angle to Find Sine of 105 degrees

Now find \( \sin(105^{\circ}) \) using \( \sin(105^{\circ}) = \cos(-15^{\circ}) \):Since the reference angle for \(105^{\circ}\) is \(15^{\circ}\), and sine and cosine are complementary for these angles in the first quadrant:\[ \sin(105^{\circ}) = \cos(15^{\circ}) \]Since \( \cos(-15^{\circ}) = \cos(15^{\circ}) \), we have:\[ \sin(105^{\circ}) = \cos(-15^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Key Concepts

Sine Function

The sine function, often written as \( \sin \), is a fundamental concept in trigonometry. It describes the ratio of the length of the opposite side to the hypotenuse in a right triangle. The function is periodic with a period of \( 360^{\circ} \) (or \( 2\pi \) radians), meaning it repeats its values every full rotation around a circle.

Understanding the sine function is vital as it helps us find unknown sides or angles in a triangle when certain values are given.

• Positive in the first and second quadrants
• Negative in the third and fourth quadrants
• Zero at \( 0^{\circ}, 180^{\circ}, \) and \( 360^{\circ} \)

In real-world applications, the sine function is used to model wave-like patterns in disciplines such as physics, engineering, and signal processing.

Angle Difference Identity

The angle difference identity is a powerful tool in trigonometry that simplifies calculations involving the cosine of an angle expressed as the difference of two other angles. The identity is expressed as:

\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]

This identity is especially useful for precisely calculating the cosine of specific angles, such as those not commonly found on the unit circle. By breaking down a complex angle into simpler, well-known angles, we can easily determine its cosine value.

For instance, calculating \( \cos(165^{\circ}) \) involves identifying it as \( \cos(210^{\circ} - 45^{\circ}) \) and applying the identity, which helps integrate known values of \( \cos 210^{\circ}, \sin 210^{\circ}, \cos 45^{\circ}, \) and \( \sin 45^{\circ} \) to find the answer.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. They are tools that simplify and solve complex trigonometric problems.

Some of the most significant identities include:

• Pythagorean Identity: \( \cos^2 \theta + \sin^2 \theta = 1 \)
• Angle Sum and Difference Identities
• Double Angle Identities
• Half Angle Identities

By applying these identities, problems can become much simpler, and expressing the relationship between functions, such as cosine and sine, becomes clear. For example, the Pythagorean identity can be used to find sine if the cosine is known by rearranging to \( \sin^2 \theta = 1 - \cos^2 \theta \).

Learning these identities often requires memorization, but understanding their derivation can enable a deeper grasp of trigonometry.

Exact Values of Trigonometric Functions

Finding the exact values of trigonometric functions, such as sine and cosine, is crucial in trigonometry. These exact values refer to the precise values of trigonometric functions for specific angles, usually related to special triangles.

For instance, angles \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \) often appear in many problems and have well-established values.

• \( \sin(30^{\circ}) = \frac{1}{2} \)
• \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \)
• \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \)

These exact values are beneficial when solving trigonometric equations, analyzing periodic patterns, or applying trigonometry in various scientific fields. Remembering these exact values and learning how to derive them using trigonometry's foundational principles is a valuable skill.

One often uses identities like the angle difference identity to calculate the exact value of non-standard angles by leveraging these familiar angles.