Understanding Trigonometry | How to Find the Value of sin(210°) using the Unit Circle and Reference Angle

sin(210°)

To find the value of sin(210°), we need to use the unit circle and the reference angle concept

To find the value of sin(210°), we need to use the unit circle and the reference angle concept.

The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate plane. It is divided into four quadrants, each with an angle measurement. The point where the terminal side of an angle intersects the unit circle determines the trigonometric values.

The reference angle is the positive acute angle formed between the terminal side of the given angle and the x-axis.

To determine sin(210°), we first identify its reference angle.

In the third quadrant (180°-270°), the terminal side of the given angle would intersect the unit circle at a point with a negative y-coordinate. The reference angle is the angle formed between this terminal side and the x-axis.

To find the reference angle, we subtract 180° from the given angle:
reference angle = 210° – 180° = 30°

Now that we have the reference angle of 30°, we can determine the value of sin(30°).

In trigonometry, sin(θ) is defined as the ratio of the side opposite the angle to the hypotenuse of a right triangle.

To form a right triangle with a reference angle of 30°, we can consider a 30-60-90 triangle. In this triangle, the side opposite the 30° angle has a length of 1 unit, and the hypotenuse has a length of 2 units.

Using the definition of sine, sin(30°) = opposite/hypotenuse = 1/2 = 0.5

Therefore, sin(210°) = sin(30°) = 0.5

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