Problem 116
Question
Will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$ \sin 60^{\circ} \sin 30^{\circ}=\frac{1}{2}\left[\cos \left(60^{\circ}-30^{\circ}\right)-\cos \left(60^{\circ}+30^{\circ}\right)\right] $$
Step-by-Step Solution
Verified Answer
Yes, the statement is true. Both sides of the equation simplify to \(\frac{\sqrt{3}}{4}\).
1Step 1: Evaluate the left side of the equation
Calculate the value of \(\sin 60^\circ \sin 30^\circ\). The sine of 60 degrees is \(\frac{\sqrt{3}}{2}\) and the sine of 30 degrees is \(\frac{1}{2}\). Multiplying these together, we obtain \(\frac{\sqrt{3}}{4}\).
2Step 2: Compute the difference in degree for cosine on right side of equation
Evaluate the difference for the cosine values, \(\cos (60^\circ - 30^\circ)\) and \(\cos (60^\circ + 30^\circ)\). This will give us \(\cos 30^\circ\) and \(\cos 90^\circ\) respectively.
3Step 3: Substitute the values of the cosine function
Substitute the exact values of the cosine function into the corresponding places. The cosine of 30 degrees is \(\frac{\sqrt{3}}{2}\) and the cosine of 90 degrees is 0. Therefore, \(\frac{1}{2} [\cos(60^\circ - 30^\circ) - \cos(60^\circ + 30^\circ)]\) simplifies to \(\frac{1}{2}[\frac{\sqrt{3}}{2} - 0]\), which simplifies to \(\frac{\sqrt{3}}{4}\).
Key Concepts
Sine and CosineExact Values of Trigonometric FunctionsTrigonometric Product-to-Sum Identities
Sine and Cosine
Understanding sine and cosine functions is fundamental in trigonometry. These functions relate the angles of a right triangle to the lengths of its sides. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse.
For example, let's consider a right triangle with a 30-degree angle. The sine of 30 degrees, written as \(\sin 30^\circ\), is equal to \(\frac{1}{2}\). Similarly, the cosine of 30 degrees, written as \(\cos 30^\circ\), is \(\frac{\sqrt{3}}{2}\). These are known as the exact values because they are precise and don't require approximation.
In practice, knowing the exact values of sine and cosine for common angles (like 30, 45, 60, and 90 degrees) is incredibly helpful. These values will recur in various trigonometric problems and understanding them can simplify computations significantly.
For example, let's consider a right triangle with a 30-degree angle. The sine of 30 degrees, written as \(\sin 30^\circ\), is equal to \(\frac{1}{2}\). Similarly, the cosine of 30 degrees, written as \(\cos 30^\circ\), is \(\frac{\sqrt{3}}{2}\). These are known as the exact values because they are precise and don't require approximation.
In practice, knowing the exact values of sine and cosine for common angles (like 30, 45, 60, and 90 degrees) is incredibly helpful. These values will recur in various trigonometric problems and understanding them can simplify computations significantly.
Exact Values of Trigonometric Functions
When dealing with trigonometric functions for certain standard angles, it's important to use exact values rather than approximations. This is because exact values retain all the precision of the mathematical relationships and help in performing algebraic manipulations which would be cumbersome with approximations.
For standard angles such as 30, 45, 60, 90 degrees, etc., the exact values are often expressed as fractions involving integers and square roots. These values typically come from geometric considerations, such as the properties of special triangles like the equilateral (for computing \(\sin 60^\circ\) and \(\cos 60^\circ\)) and the isosceles right triangle (for \(\sin 45^\circ\) and \(\cos 45^\circ\)).
Having the exact values at hand, as the solution suggests, aids in verifying identities or simplifying expressions. In the given exercise, knowing that \(\sin 30^\circ = \frac{1}{2}\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\) allows us to easily calculate their product and use these values in further trigonometric explorations.
For standard angles such as 30, 45, 60, 90 degrees, etc., the exact values are often expressed as fractions involving integers and square roots. These values typically come from geometric considerations, such as the properties of special triangles like the equilateral (for computing \(\sin 60^\circ\) and \(\cos 60^\circ\)) and the isosceles right triangle (for \(\sin 45^\circ\) and \(\cos 45^\circ\)).
Having the exact values at hand, as the solution suggests, aids in verifying identities or simplifying expressions. In the given exercise, knowing that \(\sin 30^\circ = \frac{1}{2}\) and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\) allows us to easily calculate their product and use these values in further trigonometric explorations.
Trigonometric Product-to-Sum Identities
The trigonometric product-to-sum identities are a group of formulas that express the product of trigonometric functions as sums or differences of trigonometric functions. These identities are incredibly useful because they this can simplify the process of integration, solve trigonometric equations, and prove other identities.
One of the key product-to-sum identities involves the sine function and can be expressed as follows:
\begin{align*}\sin\alpha \sin\beta &= \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]\end{align*}
Using this identity, the original exercise simplifies the left side of the equation which is a product of sines into a combination of cosines. As seen in the steps, we evaluate the \(\sin 60^\circ\) and \(\sin 30^\circ\) to find the product. Then, we use the identity to express this product as a sum of cosines, comparing it with the right side which already presents the cosines as a sum and difference within brackets. Recognizing and applying these identities can turn a perplexing problem into a simpler one, making it a vital tool in a student's trigonometry toolkit.
One of the key product-to-sum identities involves the sine function and can be expressed as follows:
\begin{align*}\sin\alpha \sin\beta &= \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]\end{align*}
Using this identity, the original exercise simplifies the left side of the equation which is a product of sines into a combination of cosines. As seen in the steps, we evaluate the \(\sin 60^\circ\) and \(\sin 30^\circ\) to find the product. Then, we use the identity to express this product as a sum of cosines, comparing it with the right side which already presents the cosines as a sum and difference within brackets. Recognizing and applying these identities can turn a perplexing problem into a simpler one, making it a vital tool in a student's trigonometry toolkit.
Other exercises in this chapter
Problem 115
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