Question

The value of \(\sin 105^{\circ}\) is: $$ \begin{array}{ll}-1 & \text { (b) } \frac{\sqrt{3}-1}{\sqrt{2}}\end{array} $$ (a) \(\frac{\sqrt{3}-1}{2 \sqrt{2}}\) (c) \(\frac{\sqrt{3}+1}{2 \sqrt{2}}\) (d) \(\frac{\sqrt{3}+1}{\sqrt{2}}\)

Short answer

Answer: \(\frac{\sqrt{3} + 1}{2 \sqrt{2}}\)

Step-by-step solution

Decompose the angle

Let's rewrite the angle \(105^{\circ}\) as the sum of two known angles, \(45^{\circ}\) and \(60^{\circ}\). So, we have: $$105^{\circ} = 45^{\circ} + 60^{\circ}.$$

Apply the addition formula for sine

Now, let's apply the addition formula for sine using these two angles: $$\sin 105^{\circ} = \sin (45^{\circ} + 60^{\circ}) = \sin 45^{\circ} \cos 60^{\circ} + \cos 45^{\circ} \sin 60^{\circ}.$$

Substitute the known values

We know the following values for sine and cosine of \(45^{\circ}\) and \(60^{\circ}\): $$\sin 45^{\circ} = \frac{1}{\sqrt{2}},\ \cos 45^{\circ} = \frac{1}{\sqrt{2}},\ \sin 60^{\circ} = \frac{\sqrt{3}}{2},\ \text{and} \ \cos 60^{\circ} = \frac{1}{2}.$$ Plug these values into the formula: $$\sin 105^{\circ} = \frac{1}{\sqrt{2}} \cdot \frac{1}{2} + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}$$

Simplify the expression

Now, let's simplify the expression to obtain the value of \(\sin 105^{\circ}\): $$\sin 105^{\circ} = \frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}}$$ Combine the terms: $$\sin 105^{\circ} = \frac{1 + \sqrt{3}}{2\sqrt{2}}.$$ Comparing this expression with the given options, we find that the value of \(\sin 105^{\circ}\) matches option (c): \(\frac{\sqrt{3} + 1}{2 \sqrt{2}}\).

Key concepts

The Sine Addition Formula

When dealing with the sine of the sum of two angles, the sine addition formula is a valuable tool. The formula allows us to break down the sine of a larger angle into its components, using known values for simpler angles. The sine addition formula is stated as:

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

This formula shows that the sine of a sum can be found by multiplying the sine of one angle by the cosine of the other angle, and vice versa, then adding those two products together. This allows us to find the sine of angles that aren't typically memorized or listed in trigonometric tables.
In our exercise, we used this formula to find \( \sin 105^{\circ} \) by writing it as the sum of \( 45^{\circ} \) and \( 60^{\circ} \), applying the formula, and then substituting known sine and cosine values for these angles.

Understanding Angle Decomposition

Angle decomposition helps simplify complex trigonometric problems by breaking down angles into sums or differences of angles we know. This approach is especially useful in trigonometry for finding exact values of uncommon angles.For example, in the exercise, \( 105^{\circ} \) is decomposed into \( 45^{\circ} + 60^{\circ} \). Both are simple, well-known angles with standard trigonometric values. This decomposition serves as the first step in simplifying the problem using the addition formula.
By rewriting the angle, we leverage pre-existing knowledge of smaller angles to analyze more complicated ones. This approach is beneficial for angles that don't have straightforward trigonometric solutions or aren't easily measured with standard tools.

Working with Known Trigonometric Values

Trigonometric functions like sine and cosine have known values at specific angles, usually multiples of \( 30^{\circ} \) and \( 45^{\circ} \). These values are vital in solving trigonometry problems that involve angle decomposition or the use of addition formulas.Let's look at the specific trigonometric values used in the solution:

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

These values are fundamental and frequently used, and memorizing them can greatly ease solving trigonometric problems.
In exercises like our example, substituting known values into formulas allows for simplification and calculation to find the precise value, transforming seemingly complicated expressions into manageable arithmetic.