Problem 19 - Algebra 2 and Trigonometry

Question

a. Find the exact value of \(\sin 120^{\circ}\) by using \(\sin \left(180^{\circ}-60^{\circ}\right)\) b. Find the exact value of \(\cos 120^{\circ}\) by using \(\cos \left(180^{\circ}-60^{\circ}\right)\) c. Find the exact value of \(\sin 75^{\circ}\) by using \(\sin \left(120^{\circ}-45^{\circ}\right)\) d. Use the value of \(\sin 75^{\circ}\) found in \(c\) to find \(\sin 105^{\circ}\) by using \(\sin \left(180^{\circ}-75^{\circ}\right)\) e. Use the value of \(\sin 75^{\circ}\) found in \(c\) to find \(\sin 285^{\circ}\) by using \(\sin \left(360^{\circ}-75^{\circ}\right)\)

Step-by-Step Solution

Answer

a) \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \). b) \( \cos 120^{\circ} = -\frac{1}{2} \). c) \( \sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \). d) \( \sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \). e) \( \sin 285^{\circ} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Evaluate sin(120°) Using Identity

To calculate \( \sin 120^{\circ} \), use the identity \( \sin(180^{\circ} - \theta) = \sin \theta \). Substitute \( \theta = 60^{\circ} \), we get \( \sin(180^{\circ} - 60^{\circ}) = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). Thus, \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \).

Step 2: Evaluate cos(120°) Using Identity

Next, calculate \( \cos 120^{\circ} \) utilizing the identity \( \cos(180^{\circ} - \theta) = -\cos \theta \). Substitute \( \theta = 60^{\circ} \), which provides \( \cos(180^{\circ} - 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2} \). Therefore, \( \cos 120^{\circ} = -\frac{1}{2} \).

Step 3: Evaluate sin(75°) Using Angle Subtraction Formula

To find \( \sin 75^{\circ} \), use the formula \( \sin(a - b) = \sin a \cos b - \cos a \sin b \) with \( a = 120^{\circ} \) and \( b = 45^{\circ} \). Compute \( \sin 75^{\circ} = \sin 120^{\circ} \cos 45^{\circ} - \cos 120^{\circ} \sin 45^{\circ} \). This evaluates to \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \left(-\frac{1}{2}\right) \cdot \frac{\sqrt{2}}{2} \), which simplifies to \( \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 4: Evaluate sin(105°) Using Identity

To find \( \sin 105^{\circ} \), apply \( \sin(180^{\circ} - \theta) = \sin \theta \). Since \( \theta = 75^{\circ} \), then \( \sin 105^{\circ} = \sin(180^{\circ} - 75^{\circ}) = \sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 5: Evaluate sin(285°) Using Identity

Finally, compute \( \sin 285^{\circ} \) using \( \sin(360^{\circ} - \theta) = -\sin \theta \). Here, \( \theta = 75^{\circ} \), thus \( \sin 285^{\circ} = -\sin(360^{\circ} - 75^{\circ}) = -\sin 75^{\circ} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

sine function

The sine function, denoted as \( \sin \), is a fundamental trigonometric function that relates the angle of a right triangle with the ratio of the length of the side opposite the angle to the hypotenuse. It is periodic, with a period of \( 360^{\circ} \) or \( 2\pi \) radians, meaning its values repeat over every full rotation.
The sine of an angle \( \theta \) can be found using unit circle values or trigonometric identities. For example, for angles \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \( 90^{\circ}, \) we have exact values easily accessible:

• \( \sin 0^{\circ} = 0 \)
• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 90^{\circ} = 1 \)

Understanding this function is essential as it forms the basis for solving various trigonometric problems, particularly those involving angle transformations.

cosine function

The cosine function, represented as \( \cos \), is another key trigonometric function. It gives the ratio of the adjacent side to the hypotenuse of a right triangle for a specified angle. Like the sine function, cosine is periodic, repeating every \( 360^{\circ} \) or \( 2\pi \) radians.
For some basic angles, the cosine values are easy to remember:

• \( \cos 0^{\circ} = 1 \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \cos 90^{\circ} = 0 \)

In problems such as finding \( \cos 120^{\circ} \), identities like \( \cos(180^{\circ} - \theta) = -\cos \theta \) become handy. Applying these identities simplifies the process of determining exact values for non-standard angles, which is vital for deeper exploration into trigonometry.

angle subtraction formula

The angle subtraction formula is invaluable in trigonometry for finding the sine and cosine of angles that are sums or differences of known angles. This formula allows us to break down complex angles into more manageable parts.
For sine, the angle subtraction formula is:\[\sin(a - b) = \sin a \cos b - \cos a \sin b\]Similarly, for cosine, the formula is:\[\cos(a - b) = \cos a \cos b + \sin a \sin b\]These formulas help in solving problems like finding \( \sin 75^{\circ} \) by expressing it as \( \sin(120^{\circ} - 45^{\circ}) \). You use known values for 120°, 45°, and then apply the formula to find a precise result.
Grasping these formulas allows you to solve a broader range of angles, extending your ability to compute exact trigonometric values without a calculator.

exact trigonometric values

Exact trigonometric values are crucial for solving trigonometry problems without relying on approximations or technology. These values derive from the unit circle and special triangle ratios.
Key angles with exact trigonometric values, such as 30°, 45°, 60°, are fundamental as they correspond to simple fractions or roots:

• For \( 30^{\circ} \) and \( 60^{\circ}, \) we use the properties of a 30-60-90 triangle, where the ratio of sides is \( 1:\sqrt{3}:2 \).
• In a 45°-45°-90° triangle, sides are proportional to \( 1:1:\sqrt{2} \), giving values like \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Understanding and memorizing these values is pivotal. They allow you to solve problems like those given in the exercise where you need to find exact sine or cosine values for composed angles such as \( 75^{\circ} \), \( 105^{\circ} \), or even \( 285^{\circ} \), by using angle subtraction and addition identities effectively.