Problem 95
Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning.After using an identity to determine the exact value of \(\sin 105^{\circ}, 1\) verified the result with a calculator.
Step-by-Step Solution
Verified Answer
The statement makes sense but is potentially misleading. Although you could verify the result obtained from the trigonometric identity with a calculator, it's important to be aware that calculators typically provide approximations, not exact values.
1Step 1: Understand the statement
The statement says: 'After using an identity to determine the exact value of \( \sin 105^{\circ} \), one verified the result with a calculator.' It should be noted that the use of identities is a mathematical approach for finding exact values, while calculators generally use algorithms that provide approximate values.
2Step 2: Evaluate the statement
The statement is technically possible. One could use the sine addition identity \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) to determine the exact value of \( \sin 105^{\circ} \). Using this identity, \( \sin 105^{\circ} = \sin ( 60^{\circ} + 45^{\circ}) = \sin 60^{\circ} \cos45^{\circ} + \cos 60^{\circ} \sin 45^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \).
3Step 3: Confirm with a calculator
One could then confirm with a calculator, which would return a value of approximately 0.9659. However, because the calculator does not give the exact value (only an approximate one), any discrepancy between the two doesn’t necessarily mean the exact value calculated using the identity is wrong.
Key Concepts
Exact ValuesSine Addition IdentityAngle Addition Formula
Exact Values
Exact values in trigonometry refer to values that are derived from certain well-known angles without approximation. These angles often include common ones like 30°, 45°, 60°, and so on, for which the trigonometric functions can be expressed as simple fractions or surds. This is in contrast to using a calculator, which provides approximate values due to its computational methods.
To find the exact value of \( \sin 105^{\circ} \), we can break it down using known angles with established values. For example:
To find the exact value of \( \sin 105^{\circ} \), we can break it down using known angles with established values. For example:
- \( \sin 30^{\circ} = \frac{1}{2} \)
- \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
- \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
Sine Addition Identity
The sine addition identity is a valuable tool in trigonometry that helps us find the sine of a sum of two angles. The identity is given by:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
This allows us to expand complex angles into simpler, known angles whose sine and cosine values are exact. For example, calculating \( \sin 105^{\circ} \) becomes manageable by expressing it as the sum of 60° and 45°, both of which have known sine and cosine values. By applying the identity:
- \( \sin 60^{\circ} \cos 45^{\circ} = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{6}}{4} \)- \( \cos 60^{\circ} \sin 45^{\circ} = \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} \)
Adding these results gives \( \sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \), which is the exact value.
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
This allows us to expand complex angles into simpler, known angles whose sine and cosine values are exact. For example, calculating \( \sin 105^{\circ} \) becomes manageable by expressing it as the sum of 60° and 45°, both of which have known sine and cosine values. By applying the identity:
- \( \sin 60^{\circ} \cos 45^{\circ} = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{6}}{4} \)- \( \cos 60^{\circ} \sin 45^{\circ} = \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} \)
Adding these results gives \( \sin 105^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4} \), which is the exact value.
Angle Addition Formula
The angle addition formula is a broader concept that includes the sine addition formula. It pertains to finding the trigonometric functions of the sum or difference of two angles. These formulas are essential for calculating trigonometric values for angles not readily available on the unit circle.
In the case of angles like 105°, which are not standard unit circle values, these formulas become especially useful. By breaking down an angle like 105° into the sum of 60° and 45°, you enable the use of these identities.
In the case of angles like 105°, which are not standard unit circle values, these formulas become especially useful. By breaking down an angle like 105° into the sum of 60° and 45°, you enable the use of these identities.
- Sine angle addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
- Cosine angle addition formula: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
Other exercises in this chapter
Problem 94
Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan ^{2} x-3 \tan x+1=0$$
View solution Problem 95
Verify each identity. $$\ln e^{\tan ^{2} x-\sec ^{2} x}=-1$$
View solution Problem 95
In Exercises \(95-98,\) graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity
View solution Problem 95
Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$7 \sin ^{2} x-1=0$$
View solution