Lesson 6.6 The Unit Circle

Intercepted Arc of a Unit Circle

s=t, meaning that the arc length of a sector of the unit circle is exactly equal to the measure of the central angle. The arc length and the angle are represented by the same real number t.

Central Angle equals...

t radians, where t can be any real number

For any real number​ t, if​ P(x,y) is a point on the unit circle corresponding to​ t, then which of the following does not accurately define a trigonometric​ function? Choose the equation that does not define a trigonometric function. A.csc t=1/y​, y≠0 B.cos t=x C.cot t=y/x​, x≠0 D.sec t=1/x​, x≠0

C.cot t=y/x​, x≠0

Circumference of a unit circle

C=2π

Which of the following statements is not​ true? Choose the correct choice below. A.A point​ (a,b) lies on the graph of the unit circle if and only if a^2+b^2=1. B.The equation of the unit circle is x^2+y^2=1. C.The unit circle is a circle centered at the origin with radius 1. D.There are infinitely many points that lie on the graph of the unit circle that have integer coordinates.

D.There are infinitely many points that lie on the graph of the unit circle that have integer coordinates.

Determining the missing coordinate of a point that lies on the graph of a unit circle in a specific quadrant Ex. Determine the missing coordinate of the point (x,7/9) that lies on the graph of the unit circle in quadrant II.

From the given​ point, y=7/9. Substitute y=7/9 into the equation of the unit circle and solve for x. The standard form of the equation of the unit circle is x^2+y^2=1. Substitute y=7/9 into the equation of the unit circle and solve for x. x^2+y^2=1 x^2+(7/9)^2=1 Simplify the power on the left side of the equation. x^2+49/81=1 Continue solving for x by isolating the​ x-term on the left side of the equation. x^2+49/81=1 x^2=1−49/81 Subtract 49/81 from both sides. x^2=81/81−49/81 Use 81/81=1 to perform subtraction. x^2=32/81 Simplify the right side of the equation. Finish solving for y by using the square root property. The square root property states that the solution to x^2=c is x= ± square root of c. x^2=32/81 Use the square root property and simplify x= ±4 square root of 2/9 Remember that quadrant II is the ​upper-left quadrant of the Cartesian plane where the sign of the​ x-coordinate of all points is negative and the sign of the​ y-coordinate of all points is positive. ​ Thererefore, the missing coordinate of the point (x,7/9) that lies on the graph of the unit circle in quadrant II is x=−4 square root of 2/9. The complete point is −4 square root of 2/9, 7/9

Finding the exact values of the 6 trig functions of t Ex. The point − square root of 11/4, square root of 5/4 lies on the graph of the unit circle and corresponds to a real number t. Find the exact values of the six trigonometric functions of t.

Let t be a real number and let P=(x,y) be the point on the unit circle that corresponds to t. For the point​ P, x=− square root of 11/4 and y= square root of 5/4. For the point on the unit circle P=(x,y) that corresponds to ​t, the sine function is defined as sint=y. Determine the value of the sine function using the definition. sint=y= square root of 5/4 For the point on the unit circle P=(x,y) that corresponds to ​t, the cosine function is defined as cost=x. Determine the value of the cosine function using the definition. cost=x=− square root of 11/4 For the point on the unit circle P=(x,y) that corresponds to ​t, if x≠0​, the tangent function is defined as tant=y/x. Use the definition to find the value of the tangent function. tant=y/x = square root of 5/4 / -square root of 11/4 = − square root of (5/11) Use the reciprocals of sin, cos, and tan for the corresponding functions csc, sec, and cot ​ Therefore, the exact values of the six trigonometric functions of t corresponding to the point −114,54 are shown below. sint= square root of5/4 csct= 4/square root of5 cost= − square root of 11/4 sect= −4/ square root of 11 tant= − square root of (5/11) cott=− square root of (11/5)

The Unit Circle

a circle with a radius of 1, centered at the origin

If t<0, then point P is obtained by rotating in which direction?

clockwise

If t>0, then point P is obtained by rotating in which direction?

counterclockwise

P(x,y)

the point on the unit circle that has an arc length of t units from the point (1, 0). The measure of the central angle (in radians) is exactly the same as the arc length, t.

Dominant Range of Sin and Cos

y=sin t (y=range [-1,1] ) (t=domain [-infinite, infinite] )