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If cosine theta equals negative square root of three over two and pi over two less than theta less than pi, what are the values of sin Θ and tan Θ?

sine theta equals negative one half, tangent theta equals square root of three over three
sine theta equals negative one half, tangent theta equals negative square root of three over three
sine theta equals one half, tangent theta equals negative square root of three over three
sine theta equals one half, tangent theta equals square root of three over three

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Answer:

The values of sinθ and tanθ are:

[tex]\sin \theta=\dfrac{1}{2}[/tex]

and

[tex]\tan \theta=-\dfrac{1}{\sqrt{3}}=-\dfrac{\sqrt{3}}{3}[/tex]

Step-by-step explanation:

It is given that:

[tex]\cos \theta=-\dfrac{\sqrt{3}}{2}[/tex]

where

[tex]\dfrac{\pi}{2}<\theta<\pi[/tex]

i.e. the angle θ lie in the second quadrant.

Also, we know that in the second quadrant sine and cosecant function is positive.

whereas cosine,secant, tangent and cotangent functions are negative.

Now, we know that:

[tex]\cos (\dfrac{5\pi}{6})=-\dfrac{\sqrt{3}}{2}[/tex]

i.e. here

[tex]\theta=\dfrac{5\pi}{6}[/tex]

Hence, we have:

[tex]\sin (\theta)=\sin (\dfrac{5\pi}{6})\\\\i.e.\\\\\sin \theta=\dfrac{1}{2}[/tex]

Also,

[tex]\tan \theta=\dfrac{\sin \theta}{\cos \theta}\\\\i.e.\\\\\tan \theta=\dfrac{\dfrac{1}{2}}{\dfrac{-\sqrt{3}}{2}}\\\\i.e.\\\\\tan \theta=\dfrac{1}{-\sqrt{3}}\\\\i.e.\\\\\tan \theta=-\dfrac{1}{\sqrt{3}}[/tex]

Now, on rationalizing we have:

[tex]\tan \theta=-\dfrac{\sqrt{3}}{3}[/tex]

Trigonometric function gives the ratio of different sides of a right-angle triangle. The value of Sine(θ)=(1/2) and Tan(θ)= -(1/√3).

What are Trigonometric functions?

The trigonometric function gives the ratio of different sides of a right-angle triangle.

[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}[/tex]

where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.

Given to us

Cosine(θ) = -(√3/2)

We know that when the value of cosine is negative the angle lies in the second quadrant, and it is measured from the negative x-axis, the value of the θ,

Cosine(θ) = -(√3/2)

Cosine(30°) = -(√3/2)

We can also right the value of θ from the positive x-axis,

Cosine(180°-30°) = -(√3/2)

Cosine(150°) = -(√3/2)

Now, the value of Sine(θ) from the positive x-axis will be written as,

Sine(150°) = (1/2)

Further, we know that the value of Tangent(Tan) in the ratio of the sine and cosine, therefore,

Tan(θ) = Sine(θ)/Cosine(θ)

Tan(150°) = Sine(150°)/Cosine(150°)

Tan(150°) = (1/2)/-(√3/2)

Tan(150°) = -(1/√3)

Hence, the value of Sine(θ)=(1/2) and Tan(θ)= -(1/√3).

Learn more about Trigonometric Functions:

https://brainly.com/question/21286835

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