Chapter 14 Day 5 Trig Functions of Any Angle. The of a circle is a portion of the of a circle. arc circumference

Chapter 14Day 5Trig Functions of Any Angle

The of a circle is a portion of the of a

circle.

arccircumference

In the unit circle, the arc length is equal to the measure of the angle formed by the initial side and the terminal side.

However, in a circle that has a radius other than 1, this isn’t the case.

The arc length of a circle is the following:

Where θ is the measure of the angle in

, and r is the length of the .

radian

radiansradius

arc length r

Find the length of the arc on a circle with the given radius that is intercepted by the central angle of the given measure.

1. 2.8 cm; 330º

2. 630 mm; 56

Find the radius of a circle in which the arc of given length is intercepted by the given central angle.

3. Arc AB: 35π cm; θ=315º

4. Arc AB:

259

in.; 300o

We can also evaluate trig functions of an angle that contains a point that isn’t necessarily on the unit circle. We just need to adjust the trig ratio for the different .

When given the coordinates of a point on the terminal side of an angle, θ, in standard position, we can evaluate the six trig functions using these rules:

radius

cosθ = sec θ =

sin θ = csc θ =

tan θ = cot θ =

Where x is the of the point, y is the of the point, and r is the

of the circle.

x-coordinate

y-coordinateradius

You will need to sketch a right triangle and use the

theorem to find the length of the radius.

Pythagorean

Find the value of the six trigonometric functions of the angle θ whose terminal side in standard position passes through the given point. If the function is not defined for the angle, state so.

5. 5, 12


6. 2, 2


Try these on your own!

7.

8.

4,2 3

0,3

We can also use the same rules when given the value of one trig function and the quadrant that it lies in. Use the given to get x, y, and/or r and then use the Pythagorean theorem to find the missing value.

Find the values of the other five trigonometric functions of the angle in the given quadrant having the given function value.

9. sin 5

13; Q III


10. cos 15

17; Q I


11. csc 2

3;Q IV


12. sec 13

2; Q IV

Find and if is the point where the terminal side of in standard position intersects the unit circle and x and y satisfy the given conditions.

13.

sin cos R x, y

4x 3y, x 0

Find and if is the point where the terminal side of in standard position intersects the unit circle and x and y satisfy the given conditions.

14.

sin cos R x, y

y x 3, x 0

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Chapter 14 Day 5 Trig Functions of Any Angle. The of a circle is a portion of the of a circle. arc circumference