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Find the coordinates of a point on the unit circle that corresponds to a complex exponential representation

Find the coordinates of a point on the unit circle that corresponds to a complex exponential representation

To find the coordinates of a point on the unit circle corresponding to eiฮธ where ฮธ=5ฯ€4, we use Eulerโ€™s formula:

eiฮธ=cosโก(ฮธ)+isinโก(ฮธ)

Substituting ฮธ=5ฯ€4:

ei5ฯ€4=cosโก(5ฯ€4)+isinโก(5ฯ€4)

From the unit circle, we know:

cosโก(5ฯ€4)=โˆ’22

sinโก(5ฯ€4)=โˆ’22

Thus, the coordinates are:

(โˆ’22,โˆ’22)

Calculate the coordinates and angles on the unit circle

Calculate the coordinates and angles on the unit circle

To find the coordinates of the point on the unit circle corresponding to an angle of 5ฯ€4 radians, we use the trigonometric functions sine and cosine. For any angle ฮธ, the coordinates are given by:

(cosโกฮธ,sinโกฮธ)

For ฮธ=5ฯ€4, we have:

cosโก(5ฯ€4)=โˆ’22

sinโก(5ฯ€4)=โˆ’22

Thus, the coordinates of the point are:

(โˆ’22,โˆ’22)

What are the sine and cosine values for the angles 30ยฐ, 45ยฐ, and 60ยฐ on the unit circle?

What are the sine and cosine values for the angles 30ยฐ, 45ยฐ, and 60ยฐ on the unit circle?

To solve for the sine and cosine values for the angles 30ยฐ, 45ยฐ, and 60ยฐ on the unit circle, we need to refer to the specific values they correspond to:

For 30ยฐ (or ฯ€/6 radians):

sinโก(30ยฐ)=12

cosโก(30ยฐ)=32

For 45ยฐ (or ฯ€/4 radians):

sinโก(45ยฐ)=22

cosโก(45ยฐ)=22

For 60ยฐ (or ฯ€/3 radians):

sinโก(60ยฐ)=32

cosโก(60ยฐ)=12

Memorizing Key Angles and Coordinates on the Unit Circle

Memorizing Key Angles and Coordinates on the Unit Circle

Tomemorizekeyanglesandcoordinatesontheunitcircle,startwiththebasicanglesindegreesandradians.Recallthattheunitcirclehasaradiusof1.

1.Identifytheangles:0ยฐ,30ยฐ,45ยฐ,60ยฐ,90ยฐ,andtheircorrespondingradianmeasures:0,ฯ€6,ฯ€4,ฯ€3,ฯ€2.

2.Learnthecoordinates:Thecoordinatesfortheseanglesare(1,0),(32,12),(22,22),(12,32),and(0,1).

3.Usesymmetry:Theunitcircleissymmetrical,soyoucanusethefirstquadranttofindcoordinatesinotherquadrantsbyconsideringthesignsofthexandycoordinates.

Find the sine and cosine values for \( \frac{\pi}{4} \) using the unit circle

Find the sine and cosine values for \( \frac{\pi}{4} \) using the unit circle

To find the sine and cosine values for ฯ€4 using the unit circle, we use the coordinates of the point corresponding to that angle on the circle.

The angle ฯ€4 is equivalent to 45 degrees. On the unit circle, the coordinates of the point at this angle are (22,22).

Therefore:

sinโก(ฯ€4)=22

cosโก(ฯ€4)=22

Calculate the exact value of tan(-ฯ€/6) using the unit circle and verify by applying trigonometric identities

Calculate the exact value of tan(-ฯ€/6) using the unit circle and verify by applying trigonometric identities

Using the unit circle, first note that โˆ’ฯ€6 is equivalent to โˆ’30โˆ˜. On the unit circle, this angle corresponds to the coordinates (32,โˆ’12).

Therefore, the value of tanโก(โˆ’ฯ€6) is given by the ratio of the y-coordinate to the x-coordinate:

tanโก(โˆ’ฯ€6)=โˆ’1232=โˆ’13=โˆ’33

Verification using trigonometric identities can be done by noting that tanโก(โˆ’x)=โˆ’tanโก(x). Hence,

tanโก(โˆ’ฯ€6)=โˆ’tanโก(ฯ€6)=โˆ’33

Find the exact values of sin(225ยฐ) and cos(225ยฐ) using the unit circle

Find the exact values of sin(225ยฐ) and cos(225ยฐ) using the unit circle

To find the exact values of sinโก(225ยฐ) and cosโก(225ยฐ), we first locate 225ยฐ on the unit circle.

225ยฐ is in the third quadrant. The reference angle is 225ยฐโ€“180ยฐ=45ยฐ. In the third quadrant, the sine and cosine of the reference angle are both negative.

Therefore, sinโก(225ยฐ)=โˆ’sinโก(45ยฐ)=โ€“22 and cosโก(225ยฐ)=โˆ’cosโก(45ยฐ)=โˆ’22.

Thus, the exact values are:

sinโก(225ยฐ)=โˆ’22

cosโก(225ยฐ)=โˆ’22

Find the measure of the angle

Find the measure of the angle

Given a circle with center O, and two points A and B on the circumference of the circle, form the angle AOB. If the radius of the circle is 5 cm and the arc AB measures 8 cm, find the measure of the angle AOB in radians and degrees.

Step 1: Find the circumference of the circle using the formula:

C=2ฯ€r

Step 2: Substitute the radius into the formula:

C=2ฯ€(5)=10ฯ€ cm

Step 3: Find the fraction of the circumference that the arc AB represents:

810ฯ€=45ฯ€

Step 4: Multiply this fraction by the total measure of the circle in radians (2ฯ€):

Angle in radians=2ฯ€ร—45ฯ€=85 radians

Step 5: To convert radians to degrees, use the fact that ฯ€ radians = 180 degrees:

Angle in degrees=85ร—180ฯ€โ‰ˆ91.56ยฐ

Find the Radius of a Circle

Find the Radius of a Circle

Given that a circle has a circumference of 31.4 units, find its radius.

We know the formula for the circumference of a circle is:

C=2ฯ€r

We can rearrange this formula to solve for the radius:

r=C2ฯ€

Substitute the given circumference value into the formula:

r=31.42ฯ€

Using the approximate value of \( \pi \approx 3.14 \), we get:

r=31.42ร—3.14=31.46.28โ‰ˆ5

So, the radius of the circle is approximately 5 units.

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