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Magazine Chapter 5: Problem 9
Find exact values for each trigonometric expression. $$ \sin 105^{\circ} $$
Short Answer
Expert verified
The exact value of \(\sin 105^{\circ}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\).
Step by step solution
01
Identify angle as a sum of known angles
The angle \(105^{\circ}\) can be expressed as the sum of two angles for which we know the exact trigonometric values. We will use the formula for the sine of a sum of angles:\[ 105^{\circ} = 60^{\circ} + 45^{\circ} \]
02
Use the sine addition formula
The sine of a sum \(a + b\) is given by:\[ \sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b \] Applying this formula to \( \sin 105^{\circ} = \sin(60^{\circ} + 45^{\circ}) \), we have:\[ \sin 105^{\circ} = \sin 60^{\circ} \cdot \cos 45^{\circ} + \cos 60^{\circ} \cdot \sin 45^{\circ} \]
03
Substitute known trigonometric values
For \(60^{\circ}\), we have \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\). For \(45^{\circ}\), we have \(\sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}\).Substituting these values into the equation, we get:\[ \sin 105^{\circ} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} \]
04
Simplify the expression
Perform the multiplication and addition:\[ \sin 105^{\circ} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \]Combine the fractions:\[ \sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \]
05
Final result
The exact value of \(\sin 105^{\circ}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sine Addition Formula
To solve problems involving angles that are not standard angles (like 0掳, 30掳, 45掳, 60掳, and 90掳), the sine addition formula is incredibly useful. The sine addition formula allows you to find the sine of the sum of two angles:
\[\sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b\]This formula works by breaking the sine of a compound angle into simpler parts that can be easily computed using known values. It becomes especially handy when you need to calculate the sine of angles like 105掳, which aren't directly found on the unit circle.
\[\sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b\]This formula works by breaking the sine of a compound angle into simpler parts that can be easily computed using known values. It becomes especially handy when you need to calculate the sine of angles like 105掳, which aren't directly found on the unit circle.
- Identify two angles that add up to the target angle (like 60掳 and 45掳 for 105掳).
- Compute the sine and cosine for these known angles.
- Plug these values into the sine addition formula to find the sine of their sum.
Exact Trigonometric Values
When you work with trigonometry, it's vital to remember the exact trigonometric values of common angles. These values come from fundamental properties of triangles and special angles, which are often memorized to make solving trigonometric problems faster and easier.
For example, for 60掳 and 45掳, the exact trigonometric values are:
For example, for 60掳 and 45掳, the exact trigonometric values are:
- \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)
- \(\cos 60^{\circ} = \frac{1}{2}\)
- \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)
- \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\)
Angle Sum Identities
The concept of angle sum identities is central to solving trigonometric expressions involving compound angles. These identities allow you to express the trigonometric function of an angle sum in terms of functions of individual angles. This process simplifies the computation significantly.
Among these, the sine, cosine, and tangent addition formulas are most frequently utilized. Specifically for sine, the identity is:
\[\sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b\]This particular identity is part of a larger family of angle sum and difference identities, which provide tools for manipulating trigonometric expressions effectively. Using these identities, you can break down complex angles into simpler components, making it easier to find exact values using known calculations.
Understanding and applying these identities not only solves specific problems but also builds a strong foundation in trigonometry, enabling you to handle more challenging angles and problems with confidence.
Among these, the sine, cosine, and tangent addition formulas are most frequently utilized. Specifically for sine, the identity is:
\[\sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b\]This particular identity is part of a larger family of angle sum and difference identities, which provide tools for manipulating trigonometric expressions effectively. Using these identities, you can break down complex angles into simpler components, making it easier to find exact values using known calculations.
Understanding and applying these identities not only solves specific problems but also builds a strong foundation in trigonometry, enabling you to handle more challenging angles and problems with confidence.
