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Find the values of sine and cosine for an angle of 45 degrees in the unit circle

Find the values of sine and cosine for an angle of 45 degrees in the unit circle

First, recall that in the unit circle, an angle of 45 degrees corresponds to ฯ€4 radians.

From trigonometric identities:

sinโกฯ€4=22

cosโกฯ€4=22

Therefore, the values are:

sinโก45โˆ˜=22

cosโก45โˆ˜=22

Find the values of sin, cos, and tan at 45 degrees on the unit circle

Find the values of sin, cos, and tan at 45 degrees on the unit circle

To find the values of sin, cos, and tan at 45โˆ˜ on the unit circle, we start by noting that 45โˆ˜ is the same as ฯ€4 radians.

The coordinates of the point on the unit circle at ฯ€4 radians are (22,22). Thus,

sinโกฯ€4=22

cosโกฯ€4=22

tanโกฯ€4=sinโกฯ€4cosโกฯ€4=2222=1

Therefore, the values are:

sinโก45โˆ˜=22

cosโก45โˆ˜=22

tanโก45โˆ˜=1

Find the cosine of a point on the unit circle in the complex plane

Find the cosine of a point on the unit circle in the complex plane

Given a point on the unit circle in the complex plane, represented by the complex number z=eiฮธ, determine the value of cosโก(ฮธ).

Since z=eiฮธ, we know that:

z=cosโก(ฮธ)+isinโก(ฮธ)

Thus, the real part of z is cosโก(ฮธ). Therefore, the value of cosโก(ฮธ) is simply the real part of z.

Hence, if z=eiฮธ=cosโก(ฮธ)+isinโก(ฮธ), then cosโก(ฮธ)=Re(z).

Find the exact values of sine, cosine, and tangent for the angle that corresponds to the point where the terminal side of angle ฮธ intersects the unit circle at (cosฮธ, sinฮธ) Given that ฮธ is in the fourth quadrant and the point on the unit circle is (1/2,

Find the exact values of sine, cosine, and tangent for the angle that corresponds to the point where the terminal side of angle ฮธ intersects the unit circle at (cosฮธ, sinฮธ) Given that ฮธ is in the fourth quadrant and the point on the unit circle is (1/2,

Given that ฮธ is in the fourth quadrant and the point on the unit circle is (12,โˆ’32), we can find the exact values of sinโกฮธ, cosโกฮธ, and tanโกฮธ.

First, we recognize that (cosโกฮธ,sinโกฮธ) directly gives us the cosine and sine values:

cosโกฮธ=12

sinโกฮธ=โˆ’32

To find tanโกฮธ, we use the identity tanโกฮธ=sinโกฮธcosโกฮธ:

tanโกฮธ=โˆ’3212

tanโกฮธ=โˆ’3

Therefore, the values are:

cosโกฮธ=12

sinโกฮธ=โˆ’32

tanโกฮธ=โˆ’3

Find the sine, cosine, and tangent of 45 degrees on the unit circle

Find the sine, cosine, and tangent of 45 degrees on the unit circle

We know that at 45โˆ˜, the coordinates on the unit circle are (22,22).

Therefore,

sinโก45โˆ˜=22

cosโก45โˆ˜=22

To find tanโก45โˆ˜, we use the identity tanโกฮธ=sinโกฮธcosโกฮธ:

tanโก45โˆ˜=sinโก45โˆ˜cosโก45โˆ˜=2222=1

Find the sine and cosine of the angle ฮธ when it equals ฯ€/4 on the unit circle

Find the sine and cosine of the angle ฮธ when it equals ฯ€/4 on the unit circle

To find the sine and cosine of the angle ฮธ=ฯ€4 on the unit circle, we use the coordinates of the point where the terminal side of the angle intersects the unit circle.

The unit circle has a radius of 1, and for ฮธ=ฯ€4, the coordinates are (22,22).

Therefore,

cosโก(ฯ€4)=22

sinโก(ฯ€4)=22

Find the Cartesian coordinates of a point on the unit circle when given an angle and a trigonometric function value

Find the Cartesian coordinates of a point on the unit circle when given an angle and a trigonometric function value

Given the angle ฮธ=7ฯ€6 on the unit circle, find the Cartesian coordinates (x,y) for the corresponding point.

Since the unit circle has a radius of 1, we use the trigonometric identities for sine and cosine:

x=cosโก(ฮธ)

y=sinโก(ฮธ)

For ฮธ=7ฯ€6:

x=cosโก(7ฯ€6)=โˆ’cosโก(ฯ€6)=โˆ’32

y=sinโก(7ฯ€6)=โˆ’sinโก(ฯ€6)=โˆ’12

Therefore, the Cartesian coordinates are:

(x,y)=(โˆ’32,โˆ’12)

Find the sine, cosine, and tangent of a point on the unit circle

Find the sine, cosine, and tangent of a point on the unit circle

For the point on the unit circle corresponding to the angle ฮธ=ฯ€4, find the sine, cosine, and tangent.

Step 1: Recognize that on the unit circle, the radius is 1.

Step 2: Use the angle ฮธ=ฯ€4.

Step 3: Find sine and cosine for ฯ€4. Since ฯ€4=45โˆ˜, sinโก(ฯ€4)=cosโก(ฯ€4)=22.

Step 4: Calculate tangent using tanโกฮธ=sinโกฮธcosโกฮธ=1.

Answers:

sinโก(ฯ€4)=22

cosโก(ฯ€4)=22

tanโก(ฯ€4)=1

Find the value of sec(ฮธ) when ฮธ = ฯ€/4 on the unit circle

Find the value of sec(ฮธ) when ฮธ = ฯ€/4 on the unit circle

To find the value of secโก(ฮธ) when ฮธ=ฯ€4 on the unit circle, we first recall that secโก(ฮธ)=1cosโก(ฮธ).

At ฮธ=ฯ€4, the cosine of ฮธ is cosโก(ฯ€4)=22.

Therefore,

secโก(ฯ€4)=1cosโก(ฯ€4)=122=22=2

Find the sine, cosine, and tangent values for a 45-degree angle on the unit circle

Find the sine, cosine, and tangent values for a 45-degree angle on the unit circle

First, we need to convert the angle from degrees to radians. Since 45โˆ˜ is in the first quadrant and corresponds to ฯ€4 radians:

45โˆ˜=ฯ€4 radians

Next, we use the unit circle values for ฯ€4. The sine and cosine values are:

sinโก(ฯ€4)=cosโก(ฯ€4)=22

The tangent value is the sine value divided by the cosine value:

tanโก(ฯ€4)=sinโก(ฯ€4)cosโก(ฯ€4)=1

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