First, let's find the exact values of the trigonometric ratios for each angle using the unit circle.
a. cos 210°
cos 210° is equal to the x-coordinate of the point on the unit circle corresponding to the angle 210°. It is important to note that 210° is in the third quadrant, where the x-coordinate is negative and the y-coordinate is negative.
To find the exact value, we can use the reference angle of 30° (which is complementary to 210°).
cos 30° = √3/2
Since in the third quadrant, the cosine is negative, we get:
cos 210° = -cos 30° = -√3/2
b. cot 120°
cot 120° = 1/tan 120°
tan 120° = tan (180° - 60°) = -tan 60° = -√3
Therefore, cot 120° = 1/(-√3) = -1/√3
c. sin 135°
sin 135° is equal to the y-coordinate of the point on the unit circle corresponding to the angle 135°.
Since 135° is in the second quadrant, both sine and negative.
sin 45° = √2/2
Therefore, sin 135° = -sin 45° = -√2/2
Now, let's add the two values:
cos 210° cot 120° + 3 sin2 135°
= (-√3/2) * (-1/√3) + 3 * (-√2/2)^2
= 1/2 - 3/2
= -1
Therefore, cos 210° cot 120° + 3 sin2 135° = -1
Evaluate each expression using the unit circle. Find the exact value. Show all work.
Write the ratio in reduced fractional form with a rationalized denominator if necessary
b. cos 210° cot 120° + 3 sin2 135°
1 answer
1 answer