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Explaining Trigonometric Ratios: Sin

By Kathleen Cantor, 20 Apr 2021

Trigonometric ratios are the functions relating to a right-angled triangle. As everyone knows, a triangle has three sides: the hypotenuse (the longest side), the perpendicular (side opposite to the angle), and the base (side adjacent to the angle).

Relationship Between Sine and Other Trigonometric Ratios

Trigonometric ratios express the sides of a right-angled triangle. There are six trigonometric ratios:

  • sine
  • cosine
  • tangent
  • cosecant
  • secant
  • cotangent

Sine, Cosine, and Tangent Ratios

Sine, cosine, and tangent ratios are the ratios of the two lengths of a right-angled triangle. The ratios represent the angles formed by the right-angled triangle hypotenuse and legs. The sine ratio is the length of a side opposite the angle it represents over the hypotenuse.

Similarly, right-angled triangles have ratios that represent their base angles. Along with sine and tangent ratios, cosine ratios represent two different sides of a right-angled triangle. Specifically, cosine ratios are a triangle's side base angle over the hypotenuse.

Alongside cosine and sine ratios, the tangent ratios are the ratios of two different sides of a right-angled triangle. The tangent ratios refer to the side's ratios opposite to the length adjacent to the angle they represent.

The other trigonometric ratios, cosecant, secant and cotangent are reciprocals to the basic trigonometric ratios: sine, cosine and tangent. The secant ratio describes the hypotenuse ratio to any side opposite to a given angle of a right-angled triangle. Cosecant represents the hypotenuse ratio to the right of a right-angled triangle, whereas the cotangent is the ratio adjacent to the opposite of a right-angled triangle.

Calculating Trigonometric Ratios: Sin

If you have a right-angled triangle, the trigonometric ratios of each of the angles that are not 90 degrees can be solved using different formulas. However, we will limit our discussion to finding sine, abbreviated as sin in trigonometric ratios.

To find sine:

Sin θ = Length of the leg opposite to the angle (O) / Length of the Hypotenuse (H)

Please note that the symbol "θ" is the Greek letter for "theta," representing angles in trigonometric ratios.

Understanding The Sine Rule

Solving a triangle entails finding the length and angles of all the sides. The sine rule applies when you are given a triangle with two angles and a side; alternatively, a triangle two sides and a non-included angle. Note that the sine rule applies to any triangle as long as one side and opposite angles are provided.

To best understand the sine rule ensure you have one of the following items:

  • Two angles and any side (AAS or SSA)
  • Two sides and an angle opposite one of the sides (SSA)
  • Three sides(SSS)
  • Two sides with their angles included( SAS)

Let's go over the actual rule. In a triangle ABC, the sides are a, b and c. So you need this information to get the sine rule formula:

  • a/SinA=b/SinB=c/SinC

As a result, the sine rule can be reciprocated this way:

  • Sin/a=SinB/b=SinC/c

Example

In a triangle, ABC, C= 102.3°, B= 28.7°, and b= 27.4 feet, find the remaining two sides.

Solution

  • A= 180°-B-C
  • A= 180°-28.7°-102.3°
  • A=49.0°

Use the sine rule to get the following:

  • a/SinA=b/SinB=c/SinC

Now use b= 27.4 to get this:

  • a=b/SinB(SinA)
  • a= 27.4/Sin28.7°(49.0)
  • a= 43.06 feet

And C:

  • C = b/SinB(SinC)
  • C= 27.4/Sin28.7(Sin102.3°)
  • C=55.75 feet

Finding the Sides Using the Sine Rule

If you want to find the length of a side of a triangle, use the formula:

  • a/sin (A) = b/ sin (B)

In this formula, the lengths are on the top. You only need two parts of the formula and also one pair of the side with its opposite angle for the formula to apply.

Finding Sides Example

Example 1

Determine the length of x in a triangle where one of the sides is 7cm, and the two are 60 and 80 degrees, respectively.

Solution

Start by writing the sine rule formula.

  • a/sin (A) = b/ sin (B)

Provide the known and unknown values in the sine rule formula.

  • x/sin(80°)=7/sin(60°)

Solve the resulting equation to find the unknown side. Give your answer to 3 significant figures.

  • x/sin(80°)=7/sin(60°)

Multiply both sides by sin(80°)

  • x=7/sin(60°) × sin(80°)
  • x=7.96( written in three significant figures)
Example 2

Find the missing side in a triangle with two angles given as 32° and 95° with one of the sides measuring 21cm.

Solution
  • a/Sin(A)= b/Sin(B)
  • p/Sin(32°) = 21/Sin(95°)
  • Multiply both sides by Sin(32°)
  • p= 21/Sin(95° * Sin(32°)
  • P= 11.2 cm

Finding Angles Using the Sine Rule

To find an angle's size, use the sine rule formula where the angles are on the top.

  • Sin(A)/a= Sin(B)/b

As mentioned earlier, you only need two parts to use the sine rule, one side, and an opposite angle.

Example 1

Work out angle n° in a triangle where one of the angles is 75°, and the two sides are 8 and 10cm.

Solution

Write out the sine rule formula for finding the angles.

  • Sin A/a =Sin B/b

Fill in the known and unknown values.

  • Sin(n°)/8=Sin(75°)/10

Multiply both sides by 8

  • Sin(n°) = sin(75°)/10 * 8
  • Sin(n°) = 0.773( three significant figures)

Use the inverse-sine function to find the angle n°.

  • n° = Sin-1(0.773)
  • =50.6°

Example 2

Find the missing angle in a triangle with two sides measuring 5.1cm and 3.6 cm if one of the angles is 100°.

Solution

  • Sin(A)/a = Sin(B)/b
  • Sin(b)/3.6= Sin(100°)/5.1

Multiply both sides by 3.6

  • Sin(b)= Sin(100)/5.1 *3.6
  • Sin(b) =0.695(3 significant figures)
  • b = Sin-1(0.695)
  • b = 44.0°

Real-Life Applications of The Sine Rule in Trigonometry

There are three practical applications where the sine ratio is applied. First, the sine ratio is essential in measuring the height of a building or a mountain. The distance of a building from the ground and the elevation angle determines the building's actual height using trigonometric functions.

Second, trigonometric ratios are applied in aviation. Aviation takes into account speed, direction, distance, and direction and speed of the wind. These different variables use the sine ratio to find solutions.

Third, trigonometric ratios are useful in oceanography. Looking at waves as large right-triangles, scientists can determine the height of waves in the ocean.

Other Uses of The Sine Rule

  • Calculus
  • Describe sound and light waves
  • Map creation
  • Satellite systems

Knowing the sine trigonometric ratios is an integral part of understanding mathematics and the world around you.

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