Problem 39
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. \(\frac{5 \pi}{2}\) radians
Step-by-Step Solution
Verified Answer
The cosine and sine of the angle \(\frac{5 \pi}{2}\) radians are 0.00 and 1.00 respectively.
1Step 1: Draw the Angle
A full circle is measured as \(2 \pi\) radians. So, for \(\frac{5 \pi}{2}\) radians, it is calculated like: \(2 \pi\) (full circle) + \(\frac{1 \pi}{2}\) (half circle) = \(\frac{5 \pi}{2}\). One will therefore draw 2 full circles and another half circle. Hence, the position ends up at the positive y-axis.
2Step 2: Calculate Cosine and Sine
For angles that end on the y-axis, the cos and sin are easily determined. Cosine is the x-coordinate of the point on the unit circle and sine is the y-coordinate. When the position ends at the positive y-axis, the point on the unit circle is (0,1). Hence, cos(\(\frac{5 \pi}{2}\))=0 and sin(\(\frac{5 \pi}{2}\))=1.
3Step 3: Rounding to Nearest Hundredth
Finally, the cosine and sine values will be rounded to the nearest hundredth, if needed. In this case, cos(\(\frac{5 \pi}{2}\)) = 0.00 and sin(\(\frac{5 \pi}{2}\)) = 1.00 as the cos and sin values were already whole numbers.
Key Concepts
Unit CircleRadiansCosineSine
Unit Circle
The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It's a fundamental tool in trigonometry that helps us understand the relationships between angles and trigonometric functions.
Each point on the unit circle corresponds to an angle, measured in radians or degrees, from the positive x-axis. The coordinates of these points are directly related to the cosine and sine of the angle. This makes it easy to visualize and calculate trigonometric functions.
Whenever you have an angle, you can think of it as an arc on the unit circle. By knowing the point where the arc ends, you can determine the cosine (x-coordinate) and sine (y-coordinate) of that angle.
Each point on the unit circle corresponds to an angle, measured in radians or degrees, from the positive x-axis. The coordinates of these points are directly related to the cosine and sine of the angle. This makes it easy to visualize and calculate trigonometric functions.
Whenever you have an angle, you can think of it as an arc on the unit circle. By knowing the point where the arc ends, you can determine the cosine (x-coordinate) and sine (y-coordinate) of that angle.
Radians
Radians are a way of measuring angles based on the radius of a circle. One full revolution around a circle is equal to \(2\pi\) radians, which is equivalent to 360 degrees. This makes radians a natural unit for measuring angles because they relate directly to the circle's radius.
A radian is defined as the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. If you imagine wrapping the radius along the circle, the angle formed is one radian.
To convert between degrees and radians, you can use the formula:
A radian is defined as the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. If you imagine wrapping the radius along the circle, the angle formed is one radian.
To convert between degrees and radians, you can use the formula:
- Degrees to radians: Multiply by \(\frac{\pi}{180}\)
- Radians to degrees: Multiply by \(\frac{180}{\pi}\)
Cosine
Cosine is one of the primary trigonometric functions. It represents the x-coordinate of a point on the unit circle for a given angle.
In the context of the unit circle, if an angle \(\theta\) is drawn from the positive x-axis, the cosine of \(\theta\) is the horizontal distance from the origin to the point on the unit circle corresponding to \(\theta\).
Cosine values range between -1 and 1:
In the context of the unit circle, if an angle \(\theta\) is drawn from the positive x-axis, the cosine of \(\theta\) is the horizontal distance from the origin to the point on the unit circle corresponding to \(\theta\).
Cosine values range between -1 and 1:
- Cosine of 0 radians (0 degrees) is 1
- Cosine of \(\frac{\pi}{2}\) radians (90 degrees) is 0
- Cosine of \(\pi\) radians (180 degrees) is -1
- Cosine of \(\frac{3\pi}{2}\) radians (270 degrees) is 0
Sine
Sine is another fundamental trigonometric function. For a given angle, sine represents the y-coordinate of the corresponding point on the unit circle.
If you picture the unit circle, sine measures the vertical distance from the origin to the point where the angle intersects the circle. Like cosine, sine values also range between -1 and 1.
Key points include:
If you picture the unit circle, sine measures the vertical distance from the origin to the point where the angle intersects the circle. Like cosine, sine values also range between -1 and 1.
Key points include:
- Sine of 0 radians (0 degrees) is 0
- Sine of \(\frac{\pi}{2}\) radians (90 degrees) is 1
- Sine of \(\pi\) radians (180 degrees) is 0
- Sine of \(\frac{3\pi}{2}\) radians (270 degrees) is -1
Other exercises in this chapter
Problem 39
How many cycles does each sine function have in the interval from 0 to 2\(\pi ?\) Find the amplitude and period of each function. $$ y=-5 \sin 2 \pi \theta $$
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A periodic function completes \(m\) cycles in \(n\) seconds. What is the period of the function? Show your work.
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Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \cot 60^{\circ} $$
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