CoMo Science

Angles & Sine Function

—

in ,

Learning Objectives:

  • The sine function relates length of the opposite side of the triangle to the longest side, which is called the hypotenuse.
  • sin(0°) = 0 since at 0° the length of the opposite side is 0.
  • sin(90°) = 1 since at 90° the length of the opposite side equals the length of the hypotenuse.
  • There are 360° in a circle.
  • Angles greater than 360° indicate more than a full rotation.
  • Angles greater than 360° can be reduced to a smaller angle by subtracting 360°.
  • Positive angles indicate rotating counterclockwise.
  • Negative angles indicate rotating clockwise.

Videos that review the basics

Sine function

This triangle, △ ABC, is a right triangle. The right angle is shown by the square at angle C. The angle that we are interested in is angle A, shown by the curved line.

The sine of ∠A is found by dividing the opposite side, a, by the hypotenuse, h. Because the hypotenuse, h, is always the longest side of the triangle. The ratio a/h will always be less than 1.

As ∠A gets smaller, the opposite side gets smaller, so that when ∠A = 0, the opposite side = 0, and therefore the sin 0° = 0.

As ∠A gets larger, the opposite side gets closer to the length of the hypotenuse, so that sin 90° = 1.

How the sine function is used

The sine function can be used to find the opposite side or the hypotenuse of a right triangle.

Example 1:

If a 100 foot rope is stretched across a river, forming a 30° angle with the bank, how wide is the river?

Solution:

Drawing a picture, we see that we are trying to find the opposite side of the triangle. 

Diagram of the rope stretched across the river.

sin 30° = opposite side / hypotenuse

We can multiple each side by the hypotenuse to get:

opposite side = 100 ft x sin 30° = 100 ft x 0.5 = 50 ft, so the river is 50 ft across.

Example 2:

A rope tied to the top of a flagpole makes a 45° angle with the ground. If the flagpole is 20 feet high, how long is the rope?

Solution:

Drawing a picture, we see that we are trying to find the hypotenuse of the triangle.

Diagram showing the flagpole and rope
Drawing a picture is very helpful when solving word problems.

sin 45° = opposite side / hypotenuse

Solving for the hypotenuse, we get the equation

hypotenuse = opposite side / sin 45° = 20 ft / 0.707 = 28.3 ft, so the rope is 28.3 ft long.

More advanced topics

Trigonometry

In addition to the sine function, there are two other functions which are frequently studied in trigonometry, the cosine and the tangent. The cosine of an angle is defined as the ratio of the adjacent side divided by the hypotenuse. The tangent of an angle is defined as the ratio of the oppositie side divided by the adjacent side.

There is a lot more to trigonometry than that. It is very useful in many fields, especially physics, engineering, and navigation.

Irrational Numbers

Irrational numbers are numbers that have an infinite number of digits to the right of the decimal point that never repeat. Most of the time, the sine of an angle is an irrational number. Typically, we only use the first few digits after the decimal place since the rest of the digits don’t make much difference.

Glossary

  • hypotenuse – the longest side of a right triangle
  • sine – the ratio of the opposite side to the hypotenuse
  • opposite side – the side of a triangle that is not part of the angle being discussed
  • adjacent side – the side of a triangle that is part of the angle being discussed

Symbols

  • ° – degrees
  • △ – triangle
  • ∠ – angle
  • □ – right angle
  • ∡ – angle of interest
  • α, β, γ, θ – angles are often labeled with these Greek letters (alpha, beta, gamma, theta, respectively)

Resources

Comments

Leave a comment