Problem 75 - Algebra and Trigonometry

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$ \sin x \cos x=\frac{\sqrt{2}}{4} $$

Step-by-Step Solution

Answer

The solutions for the equation \( \sin{x} \cos{x} = \frac{\sqrt{2}}{4} \) on the interval [0, 2pi] are \(x=\frac{\pi}{8}\) and \(x= \frac{7\pi}{8}\).

Step 1: Apply Trigonometric Identity

Recognize that the left hand side \( \sin{x} \cos{x} \) is half of the trigonometric identity for \( \sin{2x} \). Thus, the equation can be rewritten as \( \frac{1}{2} \sin{2x} = \frac{\sqrt{2}}{4} \).

Step 2: Solve for \(2x\)

Multiply the equation by 2 to simplify it. This results in \( \sin{2x} = \frac{\sqrt{2}}{2} \). Solve for \(2x\) by taking the inverse sine of both sides to get \(2x = \frac{\pi}{4}\), \(2x = \frac{7\pi}{4}\) in the range of [0,2pi] .

Step 3: Solve for \(x\)

Divide \(2x\) by 2 to solve for \(x\). The solutions within the interval [0, 2pi] are \( x = \frac{\pi}{8}\) and \(x= \frac{7\pi}{8}\).

Key Concepts

Sine Function

The sine function is an important trigonometric function that helps us understand the relationship between angles and sides in trigonometry. Its main purpose is to provide the ratio of the length of the side opposite an angle to the hypotenuse in a right-angled triangle.

• Symbol: The sine function is denoted by \( \sin \).
• Range: The values that the sine function can take lie between -1 and 1 inclusive.
• Period: The sine function repeats itself every \( 2\pi \) radians, so it has a period of \( 2\pi \).
• Waveform: The graph of the sine function is a smooth wave that starts at the origin (0, 0), peaks at \( \frac{\pi}{2} \), and repeats every cycle.

Understanding the sine function allows us to solve various trigonometric equations, including those using identities. In our exercise, the sine component of the identity connects the angle conversion to solve for values of \( x \). Recognizing when to re-express in terms of sine can simplify complex trigonometric problems, just like transforming \( \sin{x} \cos{x} \) into \( \frac{1}{2} \sin{2x} \).

Cosine Function

The cosine function is alongside sine one of the fundamental trigonometric functions. It complements the sine function by associating angles with side ratios in triangles. Specifically, cosine relates the length of the adjacent side to the hypotenuse.

• Symbol: The cosine function is denoted by \( \cos \).
• Range: The cosine function also takes values from -1 to 1 inclusive.
• Period: Like sine, cosine also has a periodicity of \( 2\pi \), meaning it repeats its values in cycles of \( 2\pi \) radians.
• Waveform: The graph of the cosine function is similar to the sine wave but starts at its maximum point when \( x=0 \).

The cosine function is crucial in deriving trigonometric identities that simplify calculations. In developing strategies for solving equations like \( \sin{x} \cos{x} \), recognizing cosine’s role helps transform equations into manageable forms by using identities such as \( \sin{2x} = 2\sin{x}\cos{x} \).

Angle Conversion

Angle conversion is a method used in trigonometry to switch between different angle units or express angles in alternate forms for computation ease. Most commonly, trigonometry operates with radians, but degrees are also frequently used.

Radians and Degrees

• Degrees are the traditional way to measure angles, where a full circle is 360°.
• Radians measure angles based on the radius of a circle, where a full circle is \( 2\pi \) radians.
• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \).
• To convert from radians to degrees, multiply by \( \frac{180}{\pi} \).

Applications in Trigonometry

Angle conversion is useful when solving problems on different domains, especially in cases where the problem spans more than one cycle of a trigonometric function’s period. For instance, reshaping the angle in the double angle identities helps convert complex trigonometric expressions into simpler forms, like when the equation for \( 2x \) is tackled in the original exercise.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle that corresponds to a particular trigonometric function value. They reverse the original trigonometric functions, thus allowing us to work backward from a function value to the angle.

• Inverse of Sine: Known as arcsine, denoted \( \sin^{-1} \) or \( \arcsin \).
• Inverse of Cosine: Known as arccosine, denoted \( \cos^{-1} \) or \( \arccos \).
• Inverse of Tangent: Known as arctangent, denoted \( \tan^{-1} \) or \( \arctan \).

How They Work

Inverse trigonometric functions are crucial when solving equations involving chiromes or angles. For example, if \( \sin{2x} = \frac{\sqrt{2}}{2} \), applying the inverse sine function finds the angle \( 2x \) itself, which in the given problem was solved as \( 2x = \frac{\pi}{4} \) and \( 2x = \frac{7\pi}{4} \).
These functions help in verifying solutions and easily converting solved angles back to their practical, meaningful measurements within the specific interval.