Problem 38 - Algebra 2

Question

Draw an angle in standard position with each given measure. Then find the values of the cosine and sine of the angle to the nearest hundredth. \(-\frac{2 \pi}{3}\) radians

Step-by-Step Solution

Answer

An angle in standard position of -2π/3 radians will give cosine value of -0.50 and sine value of -0.87 when rounded off to the nearest hundredth.

Step 1: Draw Angle in Standard Position

Start from the positive x-axis and rotate counter-clockwise if the value is positive and clockwise if the value is negative. Here, \(-\frac{2 \pi}{3}\) is negative, which implies a clockwise rotation of \(\frac{2 \pi}{3}\) radians.

Step 2: Calculate cosine and sine

Use the properties of the unit circle to find cosine and sine values. For any angle \(\theta\) in standard position with terminal side passing through a point \((x, y)\) on the unit circle, \(\cos(\theta) = x\) and \(\sin(\theta) = y\). Here, with \(-\frac{2 \pi}{3}\) radians, cosine is equal to \(-\frac{1}{2}\) and sine is equal to \(-\frac{\sqrt{3}}{2}\).

Step 3: Rounding values

Values are to be found to the nearest hundredth. So, round off the above calculated values. Value of cosine is thus \(-0.50\) and the value of sine is \(-0.87\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential when dealing with angles and triangles, especially in the context of the unit circle. They describe the relationship between the angles and the ratios of the sides of a right-angled triangle.
Here are the main trigonometric functions you need to know:

• Sine (\(\sin\theta\)): In a right triangle, it is the ratio of the side opposite the angle to the hypotenuse.
• Cosine (\(\cos\theta\)): This is the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\theta\)): It’s the ratio of the opposite side to the adjacent side, or simply \(\frac{\sin\theta}{\cos\theta}\).

In the unit circle, any point \((x, y)\) can represent \(\cos\theta\) and \(\sin\theta\), where the radius is 1. These values can be directly used to determine the position of an angle's terminal side. The unit circle helps simplify calculations and visualizes trigonometric functions as an angle sweeps through its cycle.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians divide a circle into \(2\pi\) parts. This measurement links angle measures directly to the circumference of the circle.
One radian is the angle created when the arc length is equal to the radius of the circle. Here’s how to convert between degrees and radians:

• Degrees to Radians: Use the formula \( \theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180}\).
• Radians to Degrees: Use the formula \( \theta \text{ (in degrees)} = \theta \text{ (in radians)} \times \frac{180}{\pi}\).

Radians provide a more natural way of understanding angles in trigonometry because they maintain a direct relationship with the circle's geometry. This often simplifies complex problems and calculations.

Angles in Standard Position

Angles in standard position start from the positive x-axis and sweep either clockwise or counter-clockwise based on their sign. This position helps standardize how we look at angles, making it easier to apply trigonometric functions.
Here are some guidelines to remember:

• Positive Angles: Rotate counter-clockwise from the positive x-axis.
• Negative Angles: Rotate clockwise from the positive x-axis.

The terminal side of the angle will intersect the unit circle at a point \((x, y)\), which helps find \(\cos\theta\) and \(\sin\theta\) easily. For example, with \(-\frac{2 \pi}{3}\) radians, the angle rotates clockwise, placing the terminal side in the third quadrant. Understanding these positions is crucial for solving many trigonometric problems.