Select all the correct answers.
If the measure of angle [tex]$\theta$[/tex] is [tex]$\frac{11 \pi}{6}$[/tex], which statements are true?
[tex]$\sin (\theta)=\frac{1}{2}$[/tex]
[tex]$\tan (\theta)=1$[/tex]
The measure of the reference angle is [tex]$60^{\circ}$[/tex]
[tex]$\cos (\theta)=\frac{\sqrt{3}}{2}$[/tex]
The measure of the reference angle is [tex]$30^{\circ}$[/tex]
The measure of the reference angle is [tex]$45^{\circ}$[/tex]
Answer (2)
Determine the reference angle for θ = 6 11 π , which lies in the fourth quadrant: 2 π − 6 11 π = 6 π = 3 0 ∘ .
Calculate sin ( θ ) = sin ( 6 11 π ) = − 2 1 , cos ( θ ) = cos ( 6 11 π ) = 2 3 , and tan ( θ ) = tan ( 6 11 π ) = − 3 3 .
Compare the calculated values with the given statements to identify the true ones.
The correct statements are: cos ( θ ) = 2 3 and the measure of the reference angle is 3 0 ∘ .
cos ( θ ) = 2 3 and reference angle is 3 0 ∘
Explanation
Find the reference angle We are given the angle θ = 6 11 π and asked to determine which statements about its trigonometric functions and reference angle are true. First, let's find the reference angle.
Calculate the reference angle Since 6 11 π is in the fourth quadrant, the reference angle is found by subtracting it from 2 π :
2 π − 6 11 π = 6 12 π − 6 11 π = 6 π Converting this to degrees, we have: 6 π ⋅ π 18 0 ∘ = 3 0 ∘ So, the reference angle is 3 0 ∘ .
Calculate trigonometric functions Now, let's calculate the trigonometric functions of θ = 6 11 π :
sin ( θ ) = sin ( 6 11 π ) Since 6 11 π is in the fourth quadrant, the sine is negative. Thus, sin ( 6 11 π ) = − sin ( 6 π ) = − 2 1 cos ( θ ) = cos ( 6 11 π ) Since 6 11 π is in the fourth quadrant, the cosine is positive. Thus, cos ( 6 11 π ) = cos ( 6 π ) = 2 3 tan ( θ ) = tan ( 6 11 π ) Since 6 11 π is in the fourth quadrant, the tangent is negative. Thus, tan ( 6 11 π ) = − tan ( 6 π ) = − 3 1 = − 3 3
Compare with given statements Now, let's compare our calculated values with the given statements: sin ( θ ) = 2 1 is false, since sin ( θ ) = − 2 1 .
tan ( θ ) = 1 is false, since tan ( θ ) = − 3 3 .
The measure of the reference angle is 6 0 ∘ is false, since the reference angle is 3 0 ∘ .
cos ( θ ) = 2 3 is true. The measure of the reference angle is 3 0 ∘ is true. The measure of the reference angle is 4 5 ∘ is false.
Final Answer Therefore, the correct statements are: cos ( θ ) = 2 3 The measure of the reference angle is 3 0 ∘
Examples
Understanding trigonometric functions and reference angles is crucial in fields like physics and engineering. For example, when analyzing projectile motion, you need to decompose the initial velocity into horizontal and vertical components using sine and cosine functions. Knowing the reference angle helps determine the correct signs and values of these components, allowing you to accurately predict the projectile's trajectory and range.
The true statements for θ = 6 11 π are that cos ( θ ) = 2 3 and the measure of the reference angle is 3 0 ∘ .
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