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Find the secant value of angle ฯ€/3 on the unit circle

Find the secant value of angle ฯ€/3 on the unit circle

To find the secant value, we first need to know the cosine value of the given angle on the unit circle.

The angle ฯ€3 corresponds to an angle of 60โˆ˜.

On the unit circle, the cosine of ฯ€3 is 12.

The secant is the reciprocal of the cosine:

secโก(ฯ€3)=1cosโก(ฯ€3)=112=2

Given an angle of 5ฯ€/6 radians, find the coordinates of the point on the unit circle corresponding to this angle

Given an angle of 5ฯ€/6 radians, find the coordinates of the point on the unit circle corresponding to this angle

Given an angle of 5ฯ€6 radians, we need to find the coordinates of the point on the unit circle corresponding to this angle.

First, note that 5ฯ€6 radians lies in the second quadrant. The reference angle for 5ฯ€6 is ฯ€โ€“5ฯ€6=ฯ€6.

The coordinates of the point corresponding to ฯ€6 on the unit circle are (cosโก(ฯ€6),sinโก(ฯ€6))=(32,12).

Since 5ฯ€6 is in the second quadrant, the x-coordinate is negative and the y-coordinate is positive. Thus, the coordinates of the point are (โˆ’32,12).

Find the sine of the given angle on the unit circle

Find the sine of the given angle on the unit circle

Given the angle ฮธ=30โˆ˜, we need to find sinโก(ฮธ) using the unit circle.

On the unit circle, the coordinates of the point corresponding to 30โˆ˜ are (32,12). The sine of an angle is the y-coordinate of this point. Therefore,

sinโก(30โˆ˜)=12

Understanding the Unit Circle: An Advanced Problem

Understanding the Unit Circle: An Advanced Problem

To understand the unit circle at an advanced level, consider the problem of determining the exact value of trigonometric functions given a point on the unit circle. Suppose a point P on the unit circle corresponds to an angle ฮธ. Given P=(โˆ’35,โˆ’45), find sinโกฮธ, cosโกฮธ, tanโกฮธ, and the corresponding coordinates for ฮธ+2ฯ€.

First, recall that for any point (x,y) on the unit circle:

x=cosโกฮธ,y=sinโกฮธ

Thus, for P=(โˆ’35,โˆ’45), we have:

cosโกฮธ=โˆ’35,sinโกฮธ=โˆ’45

Next, compute tanโกฮธ:

tanโกฮธ=sinโกฮธcosโกฮธ=โˆ’45โˆ’35=43

Lastly, for the angle ฮธ+2ฯ€, the coordinates remain the same since 2ฯ€ represents a full rotation around the unit circle:

Pฮธ+2ฯ€=(โˆ’35,โˆ’45)

Determine the cosine and sine values for an angle of 45 degrees on the unit circle

Determine the cosine and sine values for an angle of 45 degrees on the unit circle

To determine the cos and sin values for an angle of 45โˆ˜ on the unit circle, follow these steps:

1. Convert the angle from degrees to radians: 45โˆ˜=ฯ€4 radians.

2. On the unit circle, the coordinates of a point corresponding to an angle of ฯ€4 radians are given by (cosโก(ฯ€4),sinโก(ฯ€4)).

3. Using trigonometric values, we know:

cosโก(ฯ€4)=22

sinโก(ฯ€4)=22

Thus, the cosine and sine values for an angle of 45โˆ˜ are 22 and 22 respectively.

Determining the Position of -pi/2 on a Unit Circle

Determining the Position of -pi/2 on a Unit Circle

First, we recognize that the unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system. The angle \(-\frac{\pi}{2}\) radians corresponds to a rotation in the clockwise direction from the positive x-axis.

Since \(\frac{\pi}{2}\) radians corresponds to 90 degrees, \(-\frac{\pi}{2}\) represents a rotation of 90 degrees clockwise. On the unit circle, rotating 90 degrees clockwise from the positive x-axis brings us to the negative y-axis.

Therefore, the coordinates on the unit circle at \(-\frac{\pi}{2}\) are:

(0,โˆ’1)

Determine the values of trigonometric functions using the unit circle

Determine the values of trigonometric functions using the unit circle

To find the exact values of the trigonometric functions for the angle ฮธ=5ฯ€4 using the unit circle, follow these steps:

1. Locate the angle ฮธ=5ฯ€4 on the unit circle. This angle corresponds to 225โˆ˜, or 45โˆ˜ in the third quadrant.

2. In the third quadrant, both the sine and cosine values are negative. The reference angle is 45โˆ˜.

3. The coordinates for 45โˆ˜ are (22,22), so for 225โˆ˜ these coordinates are (โˆ’22,โˆ’22).

4. Therefore, sinโก(5ฯ€4)=โˆ’22 and cosโก(5ฯ€4)=โˆ’22.

5. The tangent function is tanโก(5ฯ€4)=sinโก(5ฯ€4)cosโก(5ฯ€4)=โˆ’22โˆ’22=1.

Unit Circle and Trigonometric Functions

Unit Circle and Trigonometric Functions

Consider the unit circle centered at the origin. Let point P have coordinates (x,y) on the unit circle such that the angle ฮธ formed by the positive x-axis and the radius to P is in the fourth quadrant. If the secant and tangent of ฮธ are given as secโกฮธ=5 and tanโกฮธ=โˆ’43, find the coordinates of point P.

Since secโกฮธ=1cosโกฮธ, we have cosโกฮธ=15. Since x2+y2=1 for any point on the unit circle:

x2+y2=1

Given that x=cosโกฮธ=15, we find y using tanโกฮธ=yx:

y=tanโกฮธโ‹…x=โˆ’43โ‹…15=โˆ’415

Therefore, the coordinates of point P are:

P=(15,โˆ’415)

Determine the coordinates of a point on the flipped unit circle given certain conditions

Determine the coordinates of a point on the flipped unit circle given certain conditions

Letโ€™s consider the unit circle equation flipped along y=x: x2+y2=1 becomes y=xโ‹…1โ€“x2. Given a point where the x-coordinate is 12, find the corresponding y-coordinate.

Since the point lies on the flipped unit circle, we have:

y=12โ‹…1โ€“(12)2

y=12โ‹…1โ€“14

y=12โ‹…34

y=12โ‹…32

y=34

Hence, the point on the flipped unit circle is (12,34).

Determine the exact values of sine, cosine, and tangent for a given angle on the unit circle

Determine the exact values of sine, cosine, and tangent for a given angle on the unit circle

Let ฮธ be an angle on the unit circle such that ฮธ=7ฯ€6. Determine the exact values of sinโก(ฮธ), cosโก(ฮธ), and tanโก(ฮธ).

The angle ฮธ=7ฯ€6 is in the third quadrant. The reference angle is ฯ€โ€“ฯ€6=ฯ€6. In the third quadrant, both sine and cosine are negative.

Therefore, sinโก(7ฯ€6)=โˆ’sinโก(ฯ€6)=โˆ’12

cosโก(7ฯ€6)=โˆ’cosโก(ฯ€6)=โˆ’32

and tanโก(7ฯ€6)=sinโก(7ฯ€6)cosโก(7ฯ€6)=โˆ’12โˆ’32=13=33

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