Answer:
a) 157
b) 113
c) 93
d) 45
e) 0
f) 85
d, e and f also have alternative answers using the other methods to find equivalent angles.
Explanation:
Sine and cosine have mirrored halves when looking at the unit circle.
For sine, you want to have the same y value, or in other words you want to flip the point you have around the y axis. For angles, this means for an angle between 0 - 180 degrees you find 180-a for the equivalent value of sine. for 180-360 you find 360-(a-180) = 540-a. Let me know if that doesn't make sense. If angles are ever above 360 or less than 0, add or subtract 360 until you are between 0 and 360
Cosine meanwhile flips over the x axis So coordinates get their x part made negative. Angles work out to a formula that works for all of them. if you start with an angle a, the corresponding angle that gets you the same cosine is 360-a. Just like for sine if the angle is greater than 360 or less than 0 add or subtract 360 till you are in the right range
For when you want sine of one angle to equal the cosine of the other, or the other way around you need to pay attention to if the first angle is in quadrant 1. if it is great! just do 90 - a where a is that first angle. Now, if a is greater than 90, this means you'll get a negative. Just add 360 and you'll get the right answer. Of course you could use the tricks from above to find a second equivalent answer.
Now lets answer some questions.
a) sin 23°= sin ??
this is sin and the angle is between 0 and 180 so we do 180-23 = 157. both are 67 away from 90 degrees. Another way to look at it.
b)sin 67°=sin ??
sine again and between 0 and 180 again so 180 - 67=113. Both 23 away from 90 degrees.
c)sin87°= sin ??
sine again, between 0 and 180 again so 180 - 87 = 93. Both 3 away from 90 degrees.
d)sin 45º = cos ??
This one almost threw me off. sine to cosine so 90-45 = 45. so the same angle is equivalent for sine and cosine.
e) sin 90°=cos ??
sine to cosine so 90 - 90 = 0, so there you go.
f) cos 5°=sin ??
Cosine to sine so 90-5 = 85