Find the angle in radians corresponding to a point on the unit circle given coordinates (x, y)
To find the angle ฮธ corresponding to the point
Given
We know that:
Therefore, the angle corresponding to the given point is:
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Find the angle in radians corresponding to a point on the unit circle given coordinates (x, y)
To find the angle ฮธ corresponding to the point
Given
We know that:
Therefore, the angle corresponding to the given point is:
Find the cosine value of the angle 120 degrees on the unit circle
Find the values of sine, cosine, and tangent for a given angle on the unit circle
To find the values of sine, cosine, and tangent for the angle ๐ =
1. Locate
2. Find the coordinates of the point on the unit circle at this angle. For
3. The cosine of the angle is the x-coordinate and the sine of the angle is the y-coordinate.
4. Tangent is given by
Therefore:
Given that \( \theta \) is an angle in the unit circle such that its terminal side passes through the point (a,b) If the line passing through (a,b) and the origin makes an angle \( \alpha \) with the x-axis, find the values of \( \sin(\alpha) \), \( \cos
Given that \( \theta \) is an angle in the unit circle such that its terminal side passes through the point \((a,b)\):
The coordinates \((a, b)\) on the unit circle imply that \(a = \cos(\theta)\) and \(b = \sin(\theta)\).
Since the line passing through \((a, b)\) and the origin makes an angle \( \alpha \) with the x-axis:
Given that \( \theta \) is in the second quadrant:
Find the sine and cosine of a 45-degree angle
To find the sine and cosine of a 45-degree angle, we can use the unit circle. A 45-degree angle corresponds to
On the unit circle, the coordinates for
We know from trigonometric identities that:
Therefore, the sine and cosine of a 45-degree angle are both
What are the coordinates of the point on the unit circle where the angle is ฯ/3?
To find the coordinates of the point on the unit circle at an angle of
For an angle
Therefore, the coordinates are:
Find the value of tan(ฯ/4) using the unit circle
To find the value of
The tangent function is defined as the ratio of the y-coordinate to the x-coordinate:
So,
Therefore, the value of
Find the trigonometric values for an angle in the unit circle
Given an angle \( \theta = \frac{5\pi}{4} \), find the values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \).
First, determine the reference angle in the unit circle. \( \theta = \frac{5\pi}{4} \) is in the third quadrant. The reference angle is \( \pi + \frac{\pi}{4} = \frac{5\pi}{4} \).
For the angle \( \frac{5\pi}{4} \):
\( \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)
\( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)
\( \tan(\frac{5\pi}{4}) = 1 \)
Therefore, the values are:
\( \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)
\( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)
\( \tan(\frac{5\pi}{4}) = 1 \)
Determining the Position of -ฯ/2 on the Unit Circle
To locate the position of
1. Start from the positive x-axis and move counterclockwise.
2. A negative angle indicates a clockwise direction.
So,
At
Since
Determine the Value of sec(ฮธ) Given the Coordinates on the Unit Circle
Given a point on the unit circle with coordinates (0.6, 0.8), determine the value of
Step 1: Recall the definition of the point on the unit circle:
Thus,
Step 2: Recall the definition of secant in terms of cosine:
Step 3: Substitute
Therefore,
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