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In this section you will:
  • Find function values for the sine and cosine of 30°  or  ( π 6 ) , 45°  or  ( π 4 ) , and 60  or  ( π 3 ) .
  • Identify the domain and range of sine and cosine functions.
  • Find reference angles.
  • Use reference angles to evaluate trigonometric functions.
Photo of a ferris wheel.
The Singapore Flyer is the world’s tallest Ferris wheel. (credit: ʺVibin JKʺ/Flickr)

Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.

Finding trigonometric functions using the unit circle

We have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in terms of the unit circle. Recall this a unit circle is a circle centered at the origin with radius 1, as shown in [link] . The angle (in radians) that t intercepts forms an arc of length s . Using the formula s = r t , and knowing that r = 1 , we see that for a unit circle, s = t .

The x- and y- axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.

For any angle t , we can label the intersection of the terminal side and the unit circle as by its coordinates, ( x , y ) . The coordinates x and y will be the outputs of the trigonometric functions f ( t ) = cos t and f ( t ) = sin t , respectively. This means x = cos  t and y = sin  t .

Graph of a circle with angle t, radius of 1, and an arc created by the angle with length s. The terminal side of the angle intersects the circle at the point (x,y).
Unit circle where the central angle is t radians

Unit circle

A unit circle    has a center at ( 0 , 0 ) and radius 1. In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle t .

Let ( x , y ) be the endpoint on the unit circle of an arc of arc length     s . The ( x , y ) coordinates of this point can be described as functions of the angle.

Defining sine and cosine functions from the unit circle

The sine function relates a real number t to the y -coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle t equals the y -value of the endpoint on the unit circle of an arc of length t . In [link] , the sine is equal to y . Like all functions, the sine function    has an input and an output. Its input is the measure of the angle; its output is the y -coordinate of the corresponding point on the unit circle.

The cosine function    of an angle t equals the x -value of the endpoint on the unit circle of an arc of length t . In [link] , the cosine is equal to x .

Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t. The terminal side of the angle intersects the circle at the point (x,y), which is equivalent to (cos t, sin t).

Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: sin t is the same as sin ( t ) and cos t is the same as cos ( t ) . Likewise, cos 2 t is a commonly used shorthand notation for ( cos ( t ) ) 2 . Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
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David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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Adjanou
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Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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