Problem 18 - Algebra 2 and Trigonometry

Question

a. Find the exact value of \(\cos 15^{\circ}\) by using \(\cos \left(45^{\circ}-30^{\circ}\right)\) b. Use the value of \(\cos 15^{\circ}\) found in a to find \(\cos 165^{\circ}\) by using \(\cos \left(180^{\circ}-15^{\circ}\right)\) c. Use the value of \(\cos 15^{\circ}\) found in a to find \(\cos 345^{\circ}\) by using \(\cos \left(360^{\circ}-15^{\circ}\right)\) d. Use \(\cos A=\sin \left(90^{\circ}-A\right)\) to find the exact value of \(\sin 75^{\circ} .\)

Step-by-Step Solution

Answer

a. \( \cos 15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \). b. \( \cos 165^{\circ} = -\frac{\sqrt{6} + \sqrt{2}}{4} \). c. \( \cos 345^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \). d. \( \sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Express \( \cos 15^{\circ} \)

Use the identity for the cosine of a difference: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \) to find \( \cos 15^{\circ} \), where \( a = 45^{\circ} \) and \( b = 30^{\circ} \).

Step 2: Identify and Plug in Known Values

Identify the known trigonometric function values: \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), and \( \sin 30^{\circ} = \frac{1}{2} \). Substitute these into the identity from the previous step.

Step 3: Calculate \( \cos 15^{\circ} \)

The expression becomes \( \cos 45^{\circ} \cos 30^{\circ} + \sin 45^{\circ} \sin 30^{\circ} = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \). Combining these results, the exact value is \( \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 4: Find \( \cos 165^{\circ} \) using \( \cos(180^{\circ} - 15^{\circ}) \)

Use the identity \( \cos(180^{\circ} - A) = -\cos A \). Since we have \( A = 15^{\circ} \), it implies \( \cos 165^{\circ} = -\cos 15^{\circ} = -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 5: Find \( \cos 345^{\circ} \) using \( \cos(360^{\circ} - 15^{\circ}) \)

Use the identity \( \cos(360^{\circ} - A) = \cos A \). Therefore, \( \cos 345^{\circ} = \cos 15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 6: Find \( \sin 75^{\circ} \) using \( \cos A = \sin(90^{\circ} - A) \)

Use the identity \( \sin(90^{\circ} - A) = \cos A \). Therefore, \( \sin 75^{\circ} = \cos 15^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Cosine Difference Identity

One of the key tools in trigonometry for determining the cosine of an angle that is the difference between two known angles is the cosine difference identity. This identity is given by:

• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

Using this formula allows us to break down complex angles into simpler ones whose trigonometric values are already known or can be easily found in trigonometric tables. For example, the problem asks us to calculate \(\cos 15^{\circ}\) by using the angles \(45^{\circ}\) and \(30^{\circ}\). By applying the cosine difference identity, students can compute these values accurately without trials or approximations, enabling them to find the exact values required. This identity is particularly useful because it shows how the cosine of the difference of two angles can be expressed in terms of the sines and cosines of both angles, creating a direct link between these functions.

Exact Trigonometric Values

Trigonometry relies heavily on having a set of known exact values for sine and cosine, especially for commonly used angles like \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\). These values include:

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

Knowing these exact trigonometric values is crucial because they serve as the building blocks for solving more complex problems in trigonometry. For instance, in the exercise, these known values are plugged into the cosine difference identity to determine \(\cos 15^{\circ}\). Moreover, the application of these values helps ensure calculations remain exact rather than approximated, which can be particularly important in fields requiring precise measurements.

Angle Subtraction in Trigonometry

Angle subtraction is a technique used in trigonometry to express the trigonometric functions of an angle in terms of the functions of two or more specific angles. This technique is handy when working with angles that aren't normally found in the main trigonometric tables. In the context of the exercise, angles such as \(15^{\circ}\), \(165^{\circ}\), and \(345^{\circ}\) can be dealt with by expressing them as the difference from angles like \(45^{\circ}\), \(180^{\circ}\), or \(360^{\circ}\) respectively.

• For example, \(\cos 15^{\circ} = \cos(45^{\circ} - 30^{\circ})\)
• \(\cos 165^{\circ} = \cos(180^{\circ} - 15^{\circ})\)
• \(\cos 345^{\circ} = \cos(360^{\circ} - 15^{\circ})\)

These expressions utilize the characteristics of symmetrical angles and their related identities to achieve exact results. This kind of trigonometric manipulation not only simplifies computations but also showcases the interconnectedness of trigonometric principles.

Sine and Cosine Relationship

The relationship between sine and cosine can often be described as complementary and is underlined by formulaic identities. One such relationship is:

• \( \cos A = \sin(90^{\circ} - A) \)

This identity illustrates how sine and cosine are interconnected through angle complements. For instance, when asked to find \(\sin 75^{\circ}\), one can easily calculate it using the cosine of its complementary angle, \(15^{\circ}\), since \(75^{\circ} = 90^{\circ} - 15^{\circ}\). This relationship is highly useful as it provides simplification through substitution, thus saving time and reducing computational errors that may arise from working with more complex calculations. Understanding this relationship helps students appreciate how these trigonometric functions interrelate and how one can be transformed into the other through complements.