Evaluate in simple surd form the following:
(1) Sin 225
(2) Cos 195
(3) Sin 345
(4) tan 195
remember your standard angles and half-angle formulas, then
sin225 = sin(180+45) = -sin45
cos195 = cos(180+15) = -cos15 = -cos(30/2)
sin345 = sin(360-15) = -sin15
tan195 = tan(180+15) = tan15 = tan(30/2)
Sure! I can help you with that. Let's evaluate each of the trigonometric functions step by step:
(1) To evaluate sin 225, we need to use the reference angle and symmetry in the unit circle. The reference angle of 225 degrees is 45 degrees, which lies in the second quadrant. In the second quadrant, the sine value is positive. Therefore, sin 225 = sin 45 = √2 / 2.
(2) To evaluate cos 195, we also need to use the reference angle and symmetry in the unit circle. The reference angle of 195 degrees is 15 degrees, which lies in the third quadrant. In the third quadrant, the cosine value is negative. Therefore, cos 195 = -cos 15.
To find cos 15, we can use the special angle formula: cos (45 - 30) = cos 45 cos 30 + sin 45 sin 30. Substituting the known values, we have cos 15 = (√2 / 2)(√3 / 2) + (√2 / 2)(1 / 2) = (√6 + √2) / 4.
So, cos 195 = -[(√6 + √2) / 4].
(3) To evaluate sin 345, we need to use the reference angle and symmetry in the unit circle. The reference angle of 345 degrees is 15 degrees, which lies in the fourth quadrant. In the fourth quadrant, the sine value is negative. Therefore, sin 345 = -sin 15.
To find sin 15, we can use the special angle formula: sin (45 - 30) = sin 45 cos 30 - cos 45 sin 30. Substituting the known values, we have sin 15 = (√2 / 2)(√3 / 2) - (√2 / 2)(1 / 2) = (√6 - √2) / 4.
So, sin 345 = -[(√6 - √2) / 4].
(4) To evaluate tan 195, we can use the identity tan x = sin x / cos x. Therefore, tan 195 = sin 195 / cos 195.
Using the values we obtained earlier, we have:
tan 195 = sin 345 / cos 195 = -[(√6 - √2) / 4] / -[(√6 + √2) / 4] = (√6 - √2) / (√6 + √2).
Hence, the evaluations in simple surd form are:
(1) sin 225 = √2 / 2,
(2) cos 195 = -[(√6 + √2) / 4],
(3) sin 345 = -[(√6 - √2) / 4],
(4) tan 195 = (√6 - √2) / (√6 + √2).
To evaluate these trigonometric functions in simple surd form, we can use the reference angles and the unit circle.
(1) Sin 225:
To determine the reference angle for 225 degrees, we subtract it from 360 degrees: 360 - 225 = 135 degrees. Since the sine function is negative in the third quadrant, we have sin 225 = -sin 135.
The reference angle for 135 degrees is 45 degrees. On the unit circle, the sine of 45 degrees is √2/2. Therefore, sin 135 = -√2/2.
(2) Cos 195:
To determine the reference angle for 195 degrees, we subtract it from 360 degrees: 360 - 195 = 165 degrees. Since the cosine function is negative in the second and third quadrants, we have cos 195 = -cos 165.
The reference angle for 165 degrees is 15 degrees. On the unit circle, the cosine of 15 degrees is √3/2. Therefore, cos 165 = -√3/2.
(3) Sin 345:
To determine the reference angle for 345 degrees, we subtract it from 360 degrees: 360 - 345 = 15 degrees. Since the sine function is positive in the first and second quadrants, we have sin 345 = sin 15.
The reference angle for 15 degrees is 15 degrees itself. On the unit circle, the sine of 15 degrees is 1/2. Therefore, sin 345 = sin 15 = 1/2.
(4) Tan 195:
To determine the reference angle for 195 degrees, we subtract it from 180 degrees: 180 - 195 = -15 degrees. Since the tangent function is negative in the second and fourth quadrants, we have tan 195 = -tan(-15).
The reference angle for -15 degrees is 15 degrees. On the unit circle, the tangent of 15 degrees is √3/3. Therefore, tan 195 = -tan(-15) = -√3/3.