Problem 84 - Precalculus

Question

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point. \(\sin \left(75^{\circ}\right)\)

Step-by-Step Solution

Answer

The exact value is \(\frac{\sqrt{6} + \sqrt{2}}{4}\), which is approximately 0.9659.

Step 1: Use the Angle Sum Formula

The sine of a sum of two angles can be found using the formula \( \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \). Here, we rewrite \( \sin(75^{\circ}) \) as \( \sin(45^{\circ} + 30^{\circ}) \).

Step 2: Apply the Known Values

Now apply the known values: \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \), \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), \( \sin(30^{\circ}) = \frac{1}{2} \), and \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \). This gives us:\[\sin(75^{\circ}) = \sin(45^{\circ}) \cos(30^{\circ}) + \cos(45^{\circ}) \sin(30^{\circ})\]Substitute the values:\[\sin(75^{\circ}) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)\]

Step 3: Simplify the Expression

Simplify the expression to get:\[\sin(75^{\circ}) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\]

Step 4: Calculate and Confirm with Calculator

Use a calculator to confirm the value. Calculate \( \sin(75^{\circ}) \) directly and also input the expression \( \frac{\sqrt{6} + \sqrt{2}}{4} \). Both should yield approximately 0.9659 when rounded to four decimal places.

Key Concepts

Angle Sum Formula

The angle sum formula is a powerful tool in trigonometry that helps us find the sine, cosine, or tangent of an angle that is the sum or difference of two known angles. Specifically, the formula for the sine of the sum of two angles, \( a \) and \( b \), is given by: \[ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \] This formula allows us to break down a complex angle into two simpler angles for which we already know trigonometric values. It is especially useful when dealing with angles that are not common on the unit circle, such as \(75^{\circ}\).

• For example, to calculate \(\sin(75^{\circ})\), we use the fact that \(75^{\circ} = 45^{\circ} + 30^{\circ}\).
• Then apply the angle sum formula to rewrite it as a combination of \(\sin(45^{\circ})\) and \(\sin(30^{\circ})\).
• This method simplifies calculations significantly and ensures the answer is more precise because it uses exact trigonometric values.

Exact Trigonometric Values

Exact trigonometric values refer to the known sine, cosine, and tangent values for specific angles, like \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\). These values appear frequently on the unit circle and are foundational in trigonometry.

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

These values derive from the symmetrical and geometric properties of a 30-60-90 triangle and a 45-45-90 triangle. Having these values at your fingertips is critical when applying the angle sum formula, as shown in the example of \(\sin(75^{\circ})\). By using these exact values, we maintain precision in our calculations and avoid the inaccuracies that might arise from using approximate or rounded numbers.

Sine Function

The sine function is one of the primary functions in trigonometry. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Importantly, the sine function is periodic and oscillates between -1 and 1. This makes it suitable for modeling waves and cycles in various scientific contexts.

• In the context of angles, the sine function is used to determine vertical coordinates on the unit circle, where the radius of the circle is 1.
• When dealing with compound angles, such as \(75^{\circ}\), the sine function can be effectively computed using identities like the angle sum formula.
• For instance, \(\sin(75^{\circ})\) can be determined using the formula \(\sin(45^{\circ} + 30^{\circ})\), resulting in a value that can be cross-verified with exact expressions.

Understanding the sine function's properties is crucial for solving trigonometric problems, as it provides insights into angle behavior and wave patterns, and facilitating the use of essential formulas like the angle sum identity.