If Sin32=t, express each of the following in terms of t.
1.Sin212
2.Cos58
3.Sin16
To express each of the following in terms of t, we can use the trigonometric identities.
1. Sin(212) = Sin(180 + 32) = Sin(180)*Cos(32) + Cos(180)*Sin(32) = 0*Cos(32) + (-1)*t = -t.
2. Cos(58) = Cos(90 - 32) = Sin(32) = t.
3. Sin(16) = Sin(32/2) = sqrt((1 - Cos(32)) / 2) = sqrt((1 - t) / 2).
To express each of the following trigonometric functions in terms of t, we need to use trigonometric identities and the relationship between different angles.
1. Sin212:
To find the value of sin(212) in terms of t, we need to utilize the relationship between angles and their corresponding trigonometric ratios. In this case, we can use the identity sin(180° + x) = -sin(x). We can rewrite 212 as 180° + 32°:
sin(212°) = sin(180° + 32°) = -sin(32°)
Since we are given that sin(32°) = t, we can express sin(212°) in terms of t:
sin(212°) = -t
2. Cos58:
To find the value of cos(58°) in terms of t, we'll use the relationship between sine and cosine functions and the Pythagorean identity. Specifically, cos(x) = √(1 - sin^2(x)).
Since we are given sin(32°) = t, we substitute this value in the equation:
cos(58°) = √(1 - sin^2(32°)) = √(1 - t^2)
Therefore, cos(58°) in terms of t is √(1 - t^2).
3. Sin16:
To express sin(16°) in terms of t, we'll use the difference formula for sines. The difference formula is sin(x - y) = sin(x)cos(y) - cos(x)sin(y).
We can rewrite 16° as the sum of 32° and -16°:
sin(16°) = sin(32° - 16°) = sin(32°)cos(16°) - cos(32°)sin(16°)
Since we know sin(32°) = t, we can rearrange the equation to solve for sin(16°):
sin(16°) + cos(32°)sin(16°) = sin(32°)cos(16°)
Factoring out sin(16°) on the left side of the equation gives:
(1 + cos(32°))sin(16°) = sin(32°)cos(16°)
Dividing both sides by (1 + cos(32°)):
sin(16°) = (sin(32°)cos(16°))/(1 + cos(32°))
So, sin(16°) in terms of t is (sin(32°)cos(16°))/(1 + cos(32°)).
1. 212 = 180 + 32
sin 212 = -sin32 = -t
2. cos 59 = sin32 = t , complementary angles
3. cos 2A = 1 - 2sin^2 A
cos 32 = 1 - 2sin^2 16
2sin^2 16 = 1 - cos32
sin^2 16 = (1 - cos32)/2
sin 16 = √(1 - cos32) / √2
but cos32 = √1 - sin^2 32) = √(1 - t^2)
son 16 = √ (1 - √(1-t^2) ) / √2