Problem 35 - Algebra 2

Question

For each angle \(\theta,\) find the values of \(\cos \theta\) and \(\sin \theta .\) Round your answers to the nearest hundredth. $$ 154^{\circ} $$

Step-by-Step Solution

Answer

The cosine of the \(154^{\circ}\) angle, rounded to the nearest hundredth, is -0.91. The sine of the \(154^{\circ}\) angle, rounded to two decimal places, is 0.40. Please note that slight variations may occur due to rounding, especially within most calculators.

Step 1: Convert degrees to radians

To solve problems involving angles, trigonometry functions in standard calculators are usually in radians, so it is essential to convert the degrees into radians first. A circle has \(360^{\circ}\) or \(2\pi\) radians. Hence, by equating these values, 1 radian is equal to \( \frac{180}{\pi}\) degree. Therefore, \(154^{\circ}\) can be converted to radians using the equation \(\frac{154 \times \pi }{180}\). Calculate this value to get the radians.

Step 2: Compute the cosine of the angle in radians

Now that the degree angle has been converted to radians, you can use cosine to find the cosine of the angle value, cos(\( \frac{154 \pi}{180}\)). Use the calculator to find the cosine of this radian angle. The final answer should be rounded to the nearest hundredth.

Step 3: Compute the sine of the angle in radians

Same as for cosine, compute the sine of the angle in radians, sin(\( \frac{154 \pi}{180}\)). Using a calculator, find the sine of this radian angle. The answer should be rounded to the nearest hundredth.

Key Concepts

Radians in Trigonometry

In trigonometry, radians are a fundamental unit of angular measurement. Unlike degrees, which divide a circle into 360 parts, radians relate the angle to the radius of a circle. The concept might be new and a bit tricky, but it is essential for calculations in trigonometry.A full circle is equivalent to \(2\pi\) radians. This means one radian is equal to \(\frac{180}{\pi}\) degrees. For example, converting \(154^\circ\) into radians involves multiplying by \(\frac{\pi}{180}\), as shown: \( \frac{154 \times \pi}{180} \approx 2.69\) radians. Understanding radians:

• Allows for easier mathematical integration and differentiation.
• Simplifies the formulae since they naturally link arc length and radius.

With practice, radians become intuitive and greatly simplify many trigonometric calculations.

Understanding Cosine

Cosine, abbreviated as cos, is a trigonometric function that represents the x-coordinate of a point on a unit circle. This circle is a powerful visual tool that aids in understanding and calculating trigonometric functions.For any angle \(\theta\), \(\cos \theta\) gives the ratio of the adjacent side to the hypotenuse in a right-angle triangle:

• Unit Circle Interpretation: Locate the angle \(\theta\) around the circle; the x-value of the corresponding point is \(\cos \theta\).
• Calculation: Using the radian value, such as \( \cos(2.69)\), and rounding it to the nearest hundredth gives an approximate value, crucial for precise calculations.

Cosine helps calculate distances and understand wave functions, among other applications.

Introduction to Sine

The sine function, often abbreviated as sin, is another core concept in trigonometry. It represents the y-coordinate on a unit circle for any given angle, providing critical insights into the behavior of different mathematical functions.Similar to cosine, \(\sin \theta\) relates the opposite side to the hypotenuse in a right-angle triangle:

• Unit Circle Interpretation: For an angle \(\theta\) on the circle, the y-value of the point is \(\sin \theta\).
• Concerned Calculations: For example, computing \( \sin(2.69) \) and rounding as needed to the nearest hundredth offers clear solutions for practical use.

Sine is vital in modeling oscillations and understanding periodic motion, making it a cornerstone in studies of waves and circular motion.