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

sin ( t ) = 7 25

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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

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
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Karim Reply
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Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
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unknown Reply
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is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
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Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
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how do you find the real and complex roots of a polynomial?
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This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
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The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
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by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
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austin Reply
the range is twice of the natural number which is the domain
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
more than 6000
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Practice Key Terms 4

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