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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

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Conney Reply
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jeric Reply
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jeric Reply
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Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Please were did you get y=mx+b from
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
thanks Tommy
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
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
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
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
what is f(x)=
Karim Reply
I don't understand
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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this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
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.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
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Ef Reply
Practice Key Terms 4

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