Problem 28
Question
Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$2-4 \sin ^{2} 15^{\circ}$$
Step-by-Step Solution
Verified Answer
The expression simplifies to \( \sqrt{3} \).
1Step 1: Recall the Double Angle Identity for Cosine
We know the identity for the cosine of a double angle: \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). This identity can be reorganized to solve for \( 2\sin^2(\theta) \) as: \( 2\sin^2(\theta) = 1 - \cos(2\theta) \).
2Step 2: Apply the Identity to the Problem
In the expression \( 2 - 4 \sin^2 15^{\circ} \), we see terms similar to the identity: \( 4\sin^2 15^{\circ} = 2 \times 2\sin^2 15^{\circ} \). Using Step 1, we have \( 2\sin^2 15^{\circ} = 1 - \cos(30^{\circ}) \), leading to \( 4\sin^2(15^{\circ}) = 2 - 2\cos(30^{\circ}) \).
3Step 3: Substitute into the Original Expression
Replace \( 4\sin^2 15^{\circ} \) in the expression with its equivalent from Step 2: \( 2 - (2 - 2\cos 30^{\circ}) \).
4Step 4: Simplify the Expression
The expression simplifies as follows: \( 2 - 2 + 2\cos 30^{\circ} = 2\cos 30^{\circ} \). Since \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), the expression becomes \( 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \).
Key Concepts
Double Angle IdentityCosine FunctionExact Trigonometric Values
Double Angle Identity
The Double Angle Identity is a fundamental tool in trigonometry. It is particularly useful when simplifying expressions or solving equations involving trigonometric functions. The identity for cosine is \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). This allows us to express the cosine of a double angle in terms of sine.
By rearranging the formula, one can solve for \( 2\sin^2(\theta) \) as follows: \( 2\sin^2(\theta) = 1 - \cos(2\theta) \). This rearrangement is critical when dealing with expressions like \( 2 - 4\sin^2 15^{\circ} \) because it helps identify how to substitute and simplify the expression into a single trigonometric function or number.
In problems dealing with specific angles, identifying and correctly applying the double angle identity brings you closer to an exact answer without the need for a calculator.
By rearranging the formula, one can solve for \( 2\sin^2(\theta) \) as follows: \( 2\sin^2(\theta) = 1 - \cos(2\theta) \). This rearrangement is critical when dealing with expressions like \( 2 - 4\sin^2 15^{\circ} \) because it helps identify how to substitute and simplify the expression into a single trigonometric function or number.
In problems dealing with specific angles, identifying and correctly applying the double angle identity brings you closer to an exact answer without the need for a calculator.
Cosine Function
The cosine function, an essential trigonometric function, is associated with the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. It is periodic, with a fundamental period of \(360^{\circ}\) or \(2\pi\) radians.
The cosine of certain standard angles such as \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) has well-known values that make calculations easier. For instance, \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \). Knowing these values allows one to swiftly compute and simplify trigonometric expressions.
When manipulating expressions like \(2 - 4 \sin^2 15^{\circ}\), our knowledge of the cosine function and its trigonometric values comes into play. Turning the expression into a form like \(2\cos(30^{\circ})\) simplifies it to \( \sqrt{3} \), thanks to the double angle identity and our understanding of the cosine function.
The cosine of certain standard angles such as \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) has well-known values that make calculations easier. For instance, \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \). Knowing these values allows one to swiftly compute and simplify trigonometric expressions.
When manipulating expressions like \(2 - 4 \sin^2 15^{\circ}\), our knowledge of the cosine function and its trigonometric values comes into play. Turning the expression into a form like \(2\cos(30^{\circ})\) simplifies it to \( \sqrt{3} \), thanks to the double angle identity and our understanding of the cosine function.
Exact Trigonometric Values
Understanding exact trigonometric values is crucial when solving trigonometric problems without a calculator. These values are derived from standard angles found in both degrees and radians. They provide precise answers for sine, cosine, and tangent functions. Familiarity with these values allows for quick calculation and simplification.
For example, exact values for cosine at standard angles include:
In our specific exercise, the recognition that \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) directly influences the final simplified result of \( \sqrt{3} \). Recognizing and using these exact values efficiently can solve problems elegantly without computational aid.
For example, exact values for cosine at standard angles include:
- \(\cos(0^{\circ}) = 1\)
- \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)
- \(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\)
- \(\cos(60^{\circ}) = \frac{1}{2}\)
- \(\cos(90^{\circ}) = 0\)
In our specific exercise, the recognition that \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) directly influences the final simplified result of \( \sqrt{3} \). Recognizing and using these exact values efficiently can solve problems elegantly without computational aid.
Other exercises in this chapter
Problem 28
Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal
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Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)
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Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(x+90^{\circ}\right)$$
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