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### Course: Algebra 2>Unit 11

Lesson 5: Graphs of sin(x), cos(x), and tan(x)

# Graph of y=sin(x)

The graph of y=sin(x) is like a wave that forever oscillates between -1 and 1, in a shape that repeats itself every 2π units. Specifically, this means that the domain of sin(x) is all real numbers, and the range is [-1,1]. See how we find the graph of y=sin(x) using the ​unit-circle definition of sin(x). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Why are imaginary numbers not included in the domain? What prevents them from 'working'?
• Because the domain (and range) must be within the cartesian plane (the x-y graph). Each axis on the cartesian plane contains all real numbers. Imaginary numbers exist "outside" of real numbers and therefore are not found in the cartesian plane. Numbers that can't be graphed are outside the domain and range. This illustration may help: http://www.icoachmath.com/image_md/Real%20Numbers1.gif
• can we get a sine curve from cosine curve,and vice versa?
• Yes!
sin x = cos(π/2 - x) = cos(x - π/2)
cos x = sin(π/2 - x) = -sin(x - π/2)
• What would the end behavior of the sine function be?
(Is there just not any end behavior since it's periodic? Or is there a special term to describe it?)
• There is no end. A sine wave will continue into infinity, unless you restrict its domain.
• Why does he only input angles in multiples of 90 degrees?
• because it's easier and he already knows what the graph looks like, so it would not matter what angles he chose. In fact, he could have picked and number for theta and the point would lie somewhere on the wavy line. Therefore, he chose the easy ones.
• Why would Pi over 2 be 90 degrees instead of 180 degrees?
• Because the circumference of a circle is `2πr`. Using the unit circle definition this would mean the circumference is `2π(1) or simply 2π`. So half a circle is `π` and a quarter circle, which would have angle of 90° is `2π/4 or simply π/2`. You bring up a good point though about how it's a bit confusing, and Sal touches on that in this video about Tau over Pi. https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/long-live-tau/v/tau-versus-pi
• sin (90 degree) = opp/hyp . I agree that opp is 1, but what is the hypotenus ?
• Yes. On the unit circle, the hypotenuse is always the radius, 1. We define the sine of the angle as the y coordinate, so at 90 degrees our coordinates are (0,1) and it equals the radius 1. Thus we get sin 90 = 1 and cos 90 is the x coordinate so it is thus defined as zero.
• Can we say that the domain of a sine function is minus infinity to plus infinity? Is that the same as saying the domain is 'all Real numbers' (at )?
• Yes, the domain of a sine function is from negative to positive infinity. The domain is written as (-∞,∞) in interval notion. It it the same thing as the domain is all real numbers.
• I'm pretty new to this so can some help me on what a "unit circle" is?
• The unit circle is used to help you find the exact values of trig functions of special angles (0°, 30°, 45°, 60°, 90° or their radian counterparts) and the multiples of those special angles. The circle uses the idea of symmetry to find the coordinates at which multiples of the special angles intersect the circle. The coordinates themselves are found using the ratios in a 30°-60°-90° triangle or a 45°-45°-90° triangle when the hypotenuse (radius of the circle) is 1.

• Without using calculator/four figure table,find cos b if sin b is 0.8
• You can find this with the Pythagorean identity, sin²x+cos²x=1. We know sin(b)=0.8, so sin²(b)=0.64, and
we have 0.64+cos²(b)=1
cos²(b)=0.36
cos(b)=±0.6

This is all we can determine without more information about which quadrant b lies in.
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