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682 Basic Ideas of Trigonometry
Lesson The Unit-Circle
Definition of
Cosine and Sine
Chapter 10
BIG IDEA Every point P on the unit circle has coordinates of the form (cos θ , sin θ ), where θ is the magnitude of a rotation that maps (1, 0) onto P.
In a right triangle, the two angles other than the right angle each have a measure between 0° and 90°. So the definitions of sine, cosine, and tangent given in Lesson 10-1 only apply to measures between 0° and 90°. However, the sine, cosine, and tangent functions can be defined for all real numbers. To define cosines and sines for all real numbers, we use rotations with center (0, 0).
MATERIALS compass, protractor, graph paper Work with a partner. Step 1 Draw a set of coordinate axes on a piece of graph paper. Let each side of a square on your coordinate grid have length 0.1 unit. With the origin as the center, use a compass to draw a circle with radius 1. Label the positive x-intercept of the circle as A 0. Your circle should look like the one below at the right. Step 2 a. With a protractor, locate the image of A 0 = (1, 0) under R 20. Label this point A 20. b. Use the grid to estimate the x- and y-coordinates of A 20. c. Use a calculator to find cos 20° and sin 20°. Step 3 a. With a protractor, locate R 40 (1, 0). Label it A 40. b. Use the grid to estimate the x- and y-coordinates of A 40. c. Use a calculator to find cos 40° and sin 40°. Step 4 a. Locate R 75 (1, 0). Label it A 75. b. Estimate the x- and y-coordinates of this point. c. Use a calculator to evaluate cos 75° and sin 75°.
ActivityActivity
y
x 0 0.5 1
1
1 0.
1
0.
A 20 A 0
y
x 0 0.5 1
1
1 0.
1
0.
A 20 A 0
Vocabulary
unit circle unit-circle definition of cosine and sine
Mental Math
Let g be a geometric sequence with the formula g (^) n = 120(0.75)n –^1. a. What is the second term of the sequence? b. If the sequence models the height in inches of a dropped ball after the nth bounce, from what height in feet was the ball dropped? c. Could this sequence model the number of people who have heard a rumor?
The Unit-Circle Definition of Cosine and Sine 683
Lesson 10-
Step 5 a. Look back at your work for Steps 2–4. What relationship do you see between the x- and y-coordinates of R θ (1, 0), cos θ , and sin θ? b. Use your answer to Step 5 Part a to estimate the values of cos 61° and sin 61° from your figure without a calculator. c. What is the relative error between your predictions in Step 5 Part b and the actual values of cos 61° and sin 61°? Were you within 3% of the actual values?
The Unit Circle, Sines, and Cosines
The circle you drew in the Activity is a unit circle. The unit circle is the circle with center at the origin and radius 1 unit. If the point (1, 0) on the circle is rotated around the origin with magnitude θ , then the image point ( x , y ) is also on the circle. The coordinates of the image point can be found using sines and cosines, as you should have discovered in the Activity.
Example 1
What are the coordinates of the image of (1, 0) under R 70? Solution Let A = (x, y) = R 70 (1, 0). In the figure at the right, OA = 1 because the radius of the unit circle is 1. Draw the segment from A to B = (x, 0). ABO is a right triangle with legs of length x and y, and hypotenuse of length 1. Now use the definitions of sine and cosine.
cos 70° = ___adj hyp =^
x__ 1 =^ x
sin 70° = ___opp hyp =^
y__ 1 =^ y The first coordinate is cos 70°, and the second coordinate is sin 70°. Thus, (x, y) = (cos 70°, sin 70°) ≈ (0.342, 0.940). That is, the image of (1, 0) under R 70 is (cos 70°, sin 70°), or about (0.342, 0.940). Check Use the Pythagorean Theorem with cos 70° and sin 70° as the lengths of the legs. Is (0.342) 2 + (0.940) 2 ≈ 1 2? 0.117 + 0.884 = 1.001 ≈ 1, so it checks.
The idea of Example 1 can be generalized to define the sine and cosine of any magnitude θ. Since any real number can be the magnitude of a rotation, this definition enlarges the domain of these trigonometric functions to be the set of all real numbers.
y
θ x 0 (1, 0)
(0, 1)
( 1, 0)
(0, 1)
( x , y ) =^ Rθ (1, 0)
y
θ x 0 (1, 0)
(0, 1)
( 1, 0)
(0, 1)
( x , y ) =^ Rθ (1, 0)
y
x 0 (1, 0)
1
A = ( x , y )
(^70) ˚ y xB
y
x 0 (1, 0)
1
A = ( x , y )
(^70) ˚ y xB
The Unit-Circle Definition of Cosine and Sine 685
Lesson 10-
Example 3
a. Find sin 630°. b. Find cos – 900°. Solution Add or subtract multiples of 360° from the argument until you obtain a value from 0° to 360°. a. 630° - 360° = 270°. So R 630 equals one complete revolution followed by a 270° rotation, and R 630 and R 270 have the same images. R 630 (1, 0) = R 270 (1, 0) = (0, – 1). So sin 630° = – 1.
b. Add 3 · 360° to – 900° to obtain a magnitude from 0° to 360°. – 900° + 3 · 360° = 180°. R– 900 = R 180. So cos – 900° = cos 180° = – 1.
Questions
COVERING THE IDEAS
- Fill in the Blanks If (1, 0) is rotated θ degrees around the origin, a. cos θ is the?^ -coordinate of its image. b. sin θ is the?^ -coordinate of its image.
- True or False The image of (1, 0) under R 23 is (sin 23°, cos 23°).
- Fill in the Blanks R 0 (1, 0) =?^ , so cos 0° =?^ and sin 0° =?^.
- Explain how to use the unit circle to find sin 180°.
In 5–7, use the unit circle to find the value.
- cos 90° 6. sin (–90°) 7. cos 270°
- If (1, 0) is rotated – 42° about the origin, what are the coordinates of its image, to the nearest thousandth?
- Fill in the Blanks a. A rotation of 540° equals a rotation of 360° followed by ? (^). b. The image of (1, 0) under R 540 is?^. c. Evaluate sin 540°.
In 10–12, suppose A = (1, 0), B = (0, 1), C = ( – 1, 0), and D = (0, – 1). Which of these points is the image of (1, 0) under the stated rotation?
- R 450 11. R 540 12. R – 720
y
x (1, 0)
(cos 630 ˚ , sin 630 ˚ ) =^ (0, 1)
y
x (1, 0)
(cos 630 ˚ , sin 630 ˚ ) =^ (0, 1)
x
y
x (cos( (^900) ˚ ), sin( (^900) ˚ )) =^ ( 1, 0)^ (1, 0) (^900) ˚
x
y
x (cos( (^900) ˚ ), sin( (^900) ˚ )) =^ ( 1, 0)^ (1, 0) (^900) ˚
y
x 0 A = (^) (1, 0)
B =^ (0, 1)
C = ( 1, 0)
D =^ (0, 1)
y
x 0 A = (^) (1, 0)
B =^ (0, 1)
C = ( 1, 0)
D =^ (0, 1)
686 Basic Ideas of Trigonometry
Chapter 10
In 13 and 14, evaluate without using a calculator.
- cos 450° and sin 450° 14. cos(–720°) and sin(–720°)
APPLYING THE MATHEMATICS
In 15–20, which letter on the figure at the right could stand for the indicated value of the trigonometric function?
- cos 80° 16. sin 80° 17. cos(–280°)
- sin 800° 19. cos 380° 20. sin(–340°)
In 21 and 22, find a solution to the equation between 0° and 360°. Then check your answer by using a calculator to approximate both sides of the equation to the nearest thousandth.
- cos 392° = cos x 22. sin(–440°) = sin y
- a. What is the largest possible value of cos θ? b. What is the smallest possible value of sin θ?
In 24 and 25, verify by substitution that the statement holds for the given value of θ.
- (cos θ ) 2 + (sin θ ) 2 = 1; θ = 7290°
- sin θ = sin(180° - θ ); θ = – 270°
REVIEW
- If an object has a parallax angle of 3° from two sites 100 meters apart, about how far away is the object? (Lesson 10-3)
- Why must the observation sites be very far apart to determine the distance to a star by parallax? (Lesson 10-3)
- A private plane flying at an altitude of 5000 feet begins its descent along a straight line to an airport 5 miles away. At what constant angle of depression does it need to descend? (Lesson 10-2)
- A submarine commander took a sighting at sea level of the aircraft carrier USS Enterprise, the tallest ship in the U.S. Navy at 250 feet. He knew that the top of the ship was about 210 feet above sea level, and he noted that the angle of elevation to the top of the mast was 4°. How far from the Enterprise was the submarine? (Lesson 10-1)
- Use the distance formula d = (^) √( x 1 - x 2 )^2 + (^) ( y 1 - y 2 )^2 to find the distance between (–2, 3) and (2, – 7). (Lesson 4-4)
EXPLORATION
- In Chapter 4, you used the Matrix Basis Theorem to develop rotation matrices for multiples of 90°. Use that theorem and the unit circle to produce a rotation matrix for any magnitude θ.
y
x 0 (1, 0)
( c , d ) ( a , b )
y
x 0 (1, 0)
( c , d ) ( a , b )
