<< Chapter < Page Chapter >> Page >

If cos ( t ) = 24 25 and t is in the fourth quadrant, find sin ( t ) .

sin ( t ) = 7 25

Got questions? Get instant answers now!

Finding sines and cosines of special angles

We have already learned some properties of the special angles, such as the conversion from radians to degrees. We can also calculate sines and cosines of the special angles using the Pythagorean Identity    and our knowledge of triangles.

Finding sines and cosines of 45° angles

First, we will look at angles of 45° or π 4 , as shown in [link] . A 45° 45° 90° triangle is an isosceles triangle, so the x- and y -coordinates of the corresponding point on the circle are the same. Because the x- and y -values are the same, the sine and cosine values will also be equal.

Graph of 45 degree angle inscribed within a circle with radius of 1. Equivalence between point (x,y) and (x,x) shown.

At t = π 4 , which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle    . This means the radius lies along the line y = x . A unit circle has a radius equal to 1. So, the right triangle formed below the line y = x has sides x and y   ( y = x ) , and a radius = 1. See [link] .

Graph of circle with pi/4 angle inscribed and a radius of 1.

From the Pythagorean Theorem we get

x 2 + y 2 = 1

Substituting y = x , we get

x 2 + x 2 = 1

Combining like terms we get

2 x 2 = 1

And solving for x , we get

x 2 = 1 2          x = ± 1 2

In quadrant I, x = 1 2 .

At t = π 4 or 45 degrees,

( x , y ) = ( x , x ) = ( 1 2 , 1 2 ) x = 1 2 , y = 1 2 cos t = 1 2 , sin t = 1 2

If we then rationalize the denominators, we get

cos t = 1 2 2 2 = 2 2 sin t = 1 2 2 2 = 2 2

Therefore, the ( x , y ) coordinates of a point on a circle of radius 1 at an angle of 45° are ( 2 2 , 2 2 ) .

Finding sines and cosines of 30° and 60° angles

Next, we will find the cosine and sine at an angle of 30° , or π 6 . First, we will draw a triangle inside a circle with one side at an angle of 30° , and another at an angle of −30° , as shown in [link] . If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger triangle will be 60° , as shown in [link] .

Graph of a circle with 30 degree angle and negative 30 degree angle inscribed to form a trangle.
Image of two 30/60/90 triangles back to back. Label for hypoteneuse r and side y.

Because all the angles are equal, the sides are also equal. The vertical line has length 2 y , and since the sides are all equal, we can also conclude that r = 2 y or y = 1 2 r . Since sin t = y ,

sin ( π 6 ) = 1 2 r

And since r = 1 in our unit circle    ,

sin ( π 6 ) = 1 2 ( 1 )              = 1 2

Using the Pythagorean Identity, we can find the cosine value.

cos 2 π 6 + sin 2 ( π 6 ) = 1      cos 2 ( π 6 ) + ( 1 2 ) 2 = 1                  cos 2 ( π 6 ) = 3 4 Use the square root property .                     cos ( π 6 ) = ± 3 ± 4 = 3 2 Since  y  is positive, choose the positive root .

The ( x , y ) coordinates for the point on a circle of radius 1 at an angle of 30° are ( 3 2 , 1 2 ) . At t = π 3 (60°), the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle, B A D , as shown in [link] . Angle A has measure 60° . At point B , we draw an angle A B C with measure of 60° . We know the angles in a triangle sum to 180° , so the measure of angle C is also 60° . Now we have an equilateral triangle. Because each side of the equilateral triangle A B C is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.

Graph of circle with an isoceles triangle inscribed.

The measure of angle A B D is 30°. So, if double, angle A B C is 60°. B D is the perpendicular bisector of A C , so it cuts A C in half. This means that A D is 1 2 the radius, or 1 2 . Notice that A D is the x -coordinate of point B , which is at the intersection of the 60° angle and the unit circle. This gives us a triangle B A D with hypotenuse of 1 and side x of length 1 2 .

Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Practice Key Terms 4

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask