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Sine is one of the three main trigonometric ratios. It’s based on the measurements of a right triangle and helps you find angle measures and distances, among other things. This lesson defines the sine function and gives examples of when it is used.

Trigonometric Functions

There are six main trigonometric functions:

  1. Sine (sin)
  2. Cosine (cos)
  3. Tangent (tan)
  4. Secant (sec)
  5. Cosecant (csc)
  6. Cotangent (cot)

These functions are used to relate the angles of a triangle with the sides of that triangle. Trigonometric functions are important when studying triangles and modeling periodic phenomena, such as waves, sound and light.

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The Sine Function

The sine function is defined as the ratio of the side of the triangle opposite the angle divided by the hypotenuse.

boat problem

The anchor’s chain is 30 m long and creates a line that is the hypotenuse of a right triangle. The angle between that hypotenuse and the ocean floor is 39 degrees. What is the depth of the anchor?Since it isn’t practical to dive down and measure how deep the anchor is, we can use a trigonometric ratio to figure it out.We know the angle the cable makes with the ocean floor and the length of the cable (hypotenuse).

To find the length of the side opposite the angle, d, we use the sine function.sin 39 = d/300.63 = d/30d = 18.9 m

The Sine Function as a Periodic Function

Trigonometric functions are called periodic functions because they repeat over a given period.

Look at the graph of the sine function:

Sine function graph, image by Geek3
Sine graph

You can see that the graph repeats itself at a distance of 2π. Therefore, we can say that the sine function has a period of 2π. Usually when looking at the sine function in this way, you don’t use degree measure, but radians.

Radian is the standard unit of angle measurement used in mathematics. A full circle is 2π radians, which is equal to 360°.

Sine and Its Relation to a Unit Circle.

In trigonometry, a unit circle is a circle centered on the origin (0,0) of a coordinate plane with a radius of 1.

Unit circle, image by Brews ohare
Image by Jim.belk

For example:Using the unit circle, determine the value of sin 7π/6?Find the point on the circle that corresponds to 7π/6 (HINT: It’s in Quadrant III in the lower left of the graph).What is the point on the edge of the circle at that point?It is (√(-3)/2, -1/2) Since -1/2 is the y-value, it is the value of sin 7π/6.So, the answer to the question is -1/2.

In this way, the unit circle can be used to find the sine or cosine of any angle, even those that are negative or over 360°

Lesson Summary

Sine is one of the main trigonometric functions. It is important for finding distances or height and can also be used to find angle measures. The unit circle is a tool that can be helpful for finding the value of sine for all types of angles, especially those that might not be readily apparent.

Learning Outcomes

When you have studied this lesson on sine, determine whether you can:

  • Remember the six main trigonometric functions
  • Recognize the sine function
  • Use the sine function to calculate a length or an angle of a triangle

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