If two angles are supplementary, then their sines are equal. Explain why. What about the cosines of supplementary angles? If you are not sure, calculate some examples.
then take
sin(180-x) , where we know x and 180-x are supplementary
= sin180cosx - cos180sinx
= 0 - (-1)sinx
= sinx
so sinx = sin(180-x)
now for the second
cos(180-x)
= cos180cosx + sin180sinx
= (-1)cosx + 0
= - cosx
and cos(180-x) = -cosx
Have you learned the expansion formulas for
sin(A-B) and cos(A-B)
that is,
sin(A-B) = sinAcosB - cosAsinB
I will have to know to determine which approach to show you
yes I have.
Well, when two angles are supplementary, it means that their sum is equal to 180 degrees. So, let's assume we have two angles A and B that are supplementary with A + B = 180 degrees.
Now, let's take the sine of angle A: sin(A). The sine of an angle is defined as the length of the side opposite the angle divided by the length of the hypotenuse in a right triangle.
Similarly, let's take the sine of angle B: sin(B). Since A and B are supplementary, we can write B = 180 - A.
So now, sin(B) = sin(180 - A). But the sine function is periodic, meaning sin(180 - A) is equal to sin(A) because their reference angles are the same.
In other words, the sines of supplementary angles are equal because their reference angles are congruent.
Now, let's move on to the cosines of supplementary angles. If you give me some specific angle values, I'd be happy to calculate the examples for you!
To understand why the sines of supplementary angles are equal, we need to recall the properties of supplementary angles. Two angles are considered supplementary if their sum is equal to 180 degrees.
Let's consider two supplementary angles, A and B. We can express their relationship as:
A + B = 180 degrees
Now, let's examine the sines of these angles:
sin(A) = sin(180 - B)
By applying the angle subtraction formula for sine, we can rewrite this as:
sin(A) = sin(180)cos(B) - cos(180)sin(B)
Now, we know that sin(180) = 0 and cos(180) = -1, so we can substitute these values into the equation:
sin(A) = 0*cos(B) - (-1)*sin(B)
sin(A) = 0 - (-1)*sin(B)
sin(A) = sin(B)
Hence, the sines of two supplementary angles are equal.
Now, let's consider the cosines of supplementary angles.
cos(A) = cos(180 - B)
By applying the angle subtraction formula for cosine, we can rewrite this as:
cos(A) = cos(180)cos(B) + sin(180)sin(B)
Similarly, we substitute the values of cos(180) and sin(180):
cos(A) = (-1)*cos(B) + 0*sin(B)
cos(A) = -cos(B)
Therefore, the cosines of supplementary angles are equal, but with opposite signs.
To verify this, let's calculate some examples:
Example 1:
Let angle A be 60 degrees. Angle B, which is supplementary to A, would be 120 degrees.
sin(60) = 0.866
sin(120) = 0.866
Hence, the sines of supplementary angles are indeed equal.
cos(60) = 0.5
cos(120) = -0.5
The cosines of supplementary angles are equal but with opposite signs.
Example 2:
Let angle A be 30 degrees. Angle B, which is supplementary to A, would be 150 degrees.
sin(30) = 0.5
sin(150) = 0.5
Again, the sines of supplementary angles are equal.
cos(30) = 0.866
cos(150) = -0.866
The cosines of supplementary angles are equal but with opposite signs.
By examining these examples, we can see that the pattern holds true for different angles as well. The sines of supplementary angles are equal, while the cosines are equal with opposite signs.