Question

How do you use special angles to find the ratios of sin 150 degrees?

Answer 1

In this question, we have to find the value of the sine of 150 degrees, sine is a trigonometric function. So to solve the question, we must know the details of trigonometric functions and how to find their values. All the trigonometric functions have a positive value in the first quadrant. In the second quadrant, the sine is positive while all the other functions are negative; in the third quadrant, the tan function is positive while all other functions are positive; and in the fourth quadrant, the cosine function is positive while all others are negative. 150 degrees is smaller than 180 degrees and greater than 90 degrees so it lies in the second quadrant, so we will write $\sin (180 - \theta ) = \sin \theta$
By using the above information, we can find out the sine of the given angle.

Complete step-by-step solution:
We know that $\sin (\pi - \theta ) = \sin \theta$
So, we get $\sin (180^\circ - 30^\circ ) = \sin 30^\circ$
We know that $\sin 30^\circ = \dfrac{1}{2}$
So, we get –
$\Rightarrow \sin 150^\circ = \dfrac{1}{2}$
Hence, the value of $\sin (150^\circ )$ using special angles is $\dfrac{1}{2}$.

Note: The trigonometry helps us to find the relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse. Sine, cosine and tangent are the main functions of the trigonometry while cosecant, secant and cotangent functions are their reciprocals respectively. We also know that the trigonometric functions are periodic, we know the value of the sine function when the angle lies between 0 and $\dfrac{\pi }{2}$ . That’s why we use the periodic property of these functions to find the value of the sine of the angles greater than $\dfrac{\pi }{2}$.