Trigonometry is a powerful mathematical tool used to solve problems involving triangles. By identifying and labeling the sides of a triangle as opposite, adjacent, and hypotenuse, one can utilize trigonometric functions to determine missing values. In this article, we will explore the basics of trigonometry and how it can be applied to solve various problems.

**Labeling Triangles**

The first step in using trigonometry is to label a triangle correctly, as depicted in the diagram below. This labeling is crucial in determining which function to use.

- Opposite - the side opposite the angle being considered
- Adjacent - the side next to the angle being considered
- Hypotenuse - the longest side across from the right angle

**SOHCAHTOA**

To remember which function to use, we can use the acronym SOHCAHTOA, which stands for:

- Sine - opposite/hypotenuse
- Cosine - adjacent/hypotenuse
- Tangent - opposite/adjacent

These equations are used to find the missing length of a side in a right-angled triangle. Here's how it works:

- Label the triangle with the appropriate sides.
- Select the correct equation based on the given information.
- Substitute the values into the equation and solve using a calculator and the appropriate function.

If one needs to find a missing angle, the inverse of these functions can also be used. The equations for finding a missing angle are:

- sin
^{-1}(opposite/hypotenuse) - cos
^{-1}(adjacent/hypotenuse) - tan
^{-1}(opposite/adjacent)

**Trigonometry in Non-Right-Angled Triangles**

Trigonometry can also be applied to solve problems involving non-right-angled triangles. In this case, the sine and cosine rules are used.

When labeling the triangle, it doesn't matter which angles are marked, as long as the opposite sides are correctly matched. The cosine rule is used to determine the length of a missing side when the other two sides and the angle between them are known. The formula is:

**Missing Side Length:** c^{2} = a^{2} + b^{2} - 2ab cos C

The cosine rule can also be used to find an angle when all side lengths are known:

**Missing Angle:** cos C = (a^{2} + b^{2} - c^{2}) / 2ab

The sine rule can be used to find the length of a missing side or the measure of a missing angle. The equations are:

**Missing Side Length:** a/sin A = b/sin B = c/sin C

**Missing Angle:** sin A/a = sin B/b = sin C/c

**Calculating the Area of a Triangle**

If the lengths of two sides and the angle between them are known, trigonometry can be used to determine the area of a triangle. The formula is:

Area = 1/2 * (a)(b) * sin C

**Using the Unit Circle**

The unit circle is a valuable tool in understanding trigonometric functions. It has a radius of 1 and coordinates of (0,0). When performing calculations, the coordinates provide the solution of (x,y).

- Trigonometry is a useful tool for solving problems involving triangles.
- The primary trigonometric functions are sine, cosine, and tangent.
- The acronym SOHCAHTOA can be used to remember the correct function to use.
- The sine and cosine rules are applied to non-right-angled triangles.
- The unit circle helps in comprehending trigonometric functions.

Trigonometry has long been associated with ancient Greece, but it also has roots in civilizations like Babylon and India. While its origins can be traced back to these ancient times, the modern version of trigonometry was developed in the 16th century by notable mathematicians such as Bartholomaeus Pitiscus and Johann Müller, also known as Regiomontanus.

One of the primary uses of trigonometry is its ability to help us find angles using various trigonometric functions. These functions, including sine (sin), cosine (cos), and tangent (tan), can be applied to a wide range of situations. Moreover, the inverse of these functions, denoted as sin-1, cos-1, and tan-1, can be employed to determine angles in a triangle.

For instance, if the lengths of two sides of a right triangle are known, the inverse sine function (sin-1) can be utilized to calculate the measure of the angle opposite the given sides. Similarly, other inverse trigonometric functions like cos-1 and tan-1 can be used to find angles in different scenarios.

In summary, trigonometry is a fundamental concept in mathematics that enables us to comprehend the relationship between angles and distances. Its practical applications are far-reaching and continue to be significant in various fields of study.

for Free

14-day free trial. Cancel anytime.

Join **20,000+** learners worldwide.

The first 14 days are on us

96% of learners report x2 faster learning

Free hands-on onboarding & support

Cancel Anytime