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Magazine Chapter 7: Problem 89
Use a half-angle identity to find the value of \(\sin 15^{\circ}\) and \(\cos 75^{\circ}\) in exact form. What do you notice?
Short Answer
Expert verified
\( \sin 15^{\circ} \) and \( \cos 75^{\circ} \) are both \( \frac{\sqrt{2 - \sqrt{3}}}{2} \), showing the complementary angle relationship.
Step by step solution
01
Recall the Half-Angle Identities
The half-angle identity for sine is \( \sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}} \). The identity for cosine is \( \cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos x}{2}} \).
02
Find \( \sin 15^{\circ} \) using a Half-Angle Identity
Since \( 15^{\circ} = \frac{30^{\circ}}{2} \), use the identity \( \sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}} \). We know \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \). Substituting into the identity gives \( \sin 15^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \).
03
Simplify \( \sin 15^{\circ} \)
Simplify \( \sin 15^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \) to find the exact value. First, calculate the numerator: \( 1 - \frac{\sqrt{3}}{2} = \frac{2}{2} - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} \). Hence, \( \sin 15^{\circ} = \sqrt{\frac{1}{2} \times \frac{2 - \sqrt{3}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \).
04
Find \( \cos 75^{\circ} \) using a Half-Angle Identity
Since \( 75^{\circ} = \frac{150^{\circ}}{2} \), use the identity \( \cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos x}{2}} \). We know \( \cos 150^{\circ} = -\frac{\sqrt{3}}{2} \). Substituting into the identity gives \( \cos 75^{\circ} = \sqrt{\frac{1 + \left(-\frac{\sqrt{3}}{2}\right)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \).
05
Simplify \( \cos 75^{\circ} \)
Simplify \( \cos 75^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \) just like we did for \( \sin 15^{\circ} \). So, \( \cos 75^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \).
06
Observe the Result
Notice that \( \sin 15^{\circ} \) and \( \cos 75^{\circ} \) have the same exact value: \( \frac{\sqrt{2 - \sqrt{3}}}{2} \), which illustrates the complementary angle relationship since \( \sin 15^{\circ} = \cos(90^{\circ} - 15^{\circ}) = \cos 75^{\circ} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Functions
Trigonometric functions are fundamental in understanding the relationships within triangles. They link angles to the ratios of two sides in right-angled triangles. The primary trigonometric functions are sine, cosine, and tangent.
- **Sine** relates the opposite side to the hypotenuse in a right triangle. - **Cosine** relates the adjacent side to the hypotenuse. - **Tangent** is the ratio of the opposite side to the adjacent side.
These functions are essential in various applications, from engineering to physics, and even in computer graphics. Trigonometric identities, such as the half-angle identities, allow us to find values of functions for specific angles. They can simplify expressions and help in solving trigonometric equations.
- **Sine** relates the opposite side to the hypotenuse in a right triangle. - **Cosine** relates the adjacent side to the hypotenuse. - **Tangent** is the ratio of the opposite side to the adjacent side.
These functions are essential in various applications, from engineering to physics, and even in computer graphics. Trigonometric identities, such as the half-angle identities, allow us to find values of functions for specific angles. They can simplify expressions and help in solving trigonometric equations.
Sine and Cosine
Sine and cosine are the backbone of trigonometry. They appear in various identities and equations that are used to simplify complex trigonometric expressions.
- **Sine of an angle** (\( \sin(x) \)) represents the ratio of the length of the side of the triangle opposite the angle to the hypotenuse.- **Cosine of an angle** (\( \cos(x) \)) represents the ratio of the length of the adjacent side to the hypotenuse.
These functions oscillate between -1 and 1 and are periodic with a period of \(360^{\circ} \) or \(2\pi \) radians. Sine and cosine can be used to express any trigonometric function, which makes them incredibly versatile in mathematics.
- **Sine of an angle** (\( \sin(x) \)) represents the ratio of the length of the side of the triangle opposite the angle to the hypotenuse.- **Cosine of an angle** (\( \cos(x) \)) represents the ratio of the length of the adjacent side to the hypotenuse.
These functions oscillate between -1 and 1 and are periodic with a period of \(360^{\circ} \) or \(2\pi \) radians. Sine and cosine can be used to express any trigonometric function, which makes them incredibly versatile in mathematics.
Complementary Angles
Complementary angles are pairs of angles that add up to \(90^{\circ} \). This concept is very useful in trigonometry as it helps relate different trigonometric functions. For example, the angles \(15^{\circ} \) and \(75^{\circ} \) are complementary because their sum is \(90^{\circ} \).
In trigonometry, the sine of an angle is equal to the cosine of its complementary angle:
\[\sin(x) = \cos(90^{\circ} - x)\]
\[\cos(x) = \sin(90^{\circ} - x)\]
This identity is powerful for solving problems, as seen in the exercise where \( \sin 15^{\circ} \) equates to \( \cos 75^{\circ} \).
In trigonometry, the sine of an angle is equal to the cosine of its complementary angle:
\[\sin(x) = \cos(90^{\circ} - x)\]
\[\cos(x) = \sin(90^{\circ} - x)\]
This identity is powerful for solving problems, as seen in the exercise where \( \sin 15^{\circ} \) equates to \( \cos 75^{\circ} \).
Exact Trigonometric Values
Exact trigonometric values are precise values obtained for specific angles without using a calculator. These values often involve square roots and fractions, which give us precise mathematical expressions.
Using half-angle identities, we can find exact values for angles like \(15^{\circ}\) and \(75^{\circ}\). For instance, utilizing the formulas:
\[\sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}}\]
\[\cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos x}{2}}\]
These allow us to determine exact trigonometric values for angles not commonly found on the unit circle. Calculating them helps reinforce a deeper understanding of angles and their measurements.
Using half-angle identities, we can find exact values for angles like \(15^{\circ}\) and \(75^{\circ}\). For instance, utilizing the formulas:
\[\sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}}\]
\[\cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos x}{2}}\]
These allow us to determine exact trigonometric values for angles not commonly found on the unit circle. Calculating them helps reinforce a deeper understanding of angles and their measurements.
