Problem 27 - Algebra 2 and Trigonometry

Question

In \(13-24,\) find, to the nearest ten-thousandth, the radian measure \(\theta\) of a first-quadrant angle with the given function value. If \(\mathrm{f}(x)=\sin 2 x+\cos 3 x,\) find \(\mathrm{f}\left(\frac{\pi}{4}\right)\)

Step-by-Step Solution

Answer

\( \mathrm{f}\left(\frac{\pi}{4}\right) \approx 0.2929 \)

Step 1: Substitute the Value

We begin by substituting \( x = \frac{\pi}{4} \) into the function \( \mathrm{f}(x) = \sin 2x + \cos 3x \). This gives us \( \mathrm{f}\left(\frac{\pi}{4}\right) = \sin\left(2 \times \frac{\pi}{4}\right) + \cos\left(3 \times \frac{\pi}{4}\right) \).

Step 2: Simplify the Angle Expressions

Simplify the angles in the trigonometric functions. \( 2 \times \frac{\pi}{4} = \frac{\pi}{2} \), and \( 3 \times \frac{\pi}{4} = \frac{3\pi}{4} \). So the expression becomes \( \sin\left(\frac{\pi}{2}\right) + \cos\left(\frac{3\pi}{4}\right) \).

Step 3: Evaluate the Trigonometric Functions

Calculate the sine and cosine values. The sine of \( \frac{\pi}{2} \) is \( 1 \) because \( \sin\left(\frac{\pi}{2}\right) = 1 \). The cosine of \( \frac{3\pi}{4} \) is \( -\frac{\sqrt{2}}{2} \) because \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

Step 4: Sum the Values

Combine the results from Step 3. The sum is \( 1 + \left(-\frac{\sqrt{2}}{2}\right) = 1 - \frac{\sqrt{2}}{2} \).

Step 5: Approximate to the Nearest Ten-Thousandth

Evaluate \( 1 - \frac{\sqrt{2}}{2} \). First, calculate \( \sqrt{2} \approx 1.4142 \). Then, \( \frac{\sqrt{2}}{2} \approx 0.7071 \). Therefore, \( 1 - 0.7071 = 0.2929 \).

Key Concepts

Radian Measure

The concept of radian measure is important for understanding angles in trigonometry. A radian is a way to measure angles using the radius of a circle. Imagine wrapping the radius around the circle's edge. When the angle matches the arc length equal to the radius, it's one radian. This is a natural way to measure angles because it connects the angle directly with the size of the circle.To convert from degrees to radians, use the relationship that 180 degrees is equivalent to \\( \pi \) radians. Hence:

• To convert degrees to radians: multiply the degrees by \( \frac{\pi}{180} \).
• To convert radians to degrees: multiply the radians by \( \frac{180}{\pi} \).

Radian measure simplifies many trigonometric calculations, especially when evaluating functions like sine and cosine where the periodicity and symmetry appear more naturally.

First-Quadrant Angles

First-quadrant angles are those between 0 and \( \frac{\pi}{2} \) radians (0 to 90 degrees). In this region, both sine and cosine values are positive. Understanding the behavior of trigonometric functions within the first quadrant is crucial for solving various trigonometric problems and it helps to predict values without directly calculating them.Here's why first-quadrant angles are useful:

• They serve as a reference: Values of sine and cosine for these angles (0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \)) are often memorized as they frequently appear in calculations.
• They simplify problem-solving: When angles are outside the first quadrant, they can often be expressed in terms of first-quadrant angles due to symmetry.

Knowing the signs and approximate values of trigonometric functions in the first quadrant lays a foundation for understanding these functions across all quadrants.

Trigonometric Identities

Trigonometric identities are equations that hold true for all angle values. They are useful tools for simplifying expressions and solving trigonometric equations. Common identities include the Pythagorean identity, angle sum and difference identities, and double angle identities.Understanding some key identities:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Double Angle Formulas: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) and \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• Sum and Difference Formulas: \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)

These identities assist in calculating complex trigonometric expressions and verifying solutions. They also make it easier to remember key values and behaviors of trigonometric functions.