Trigonometry is the branch of mathematics that deals with the study of angles and their numerical calculations. Calculators come in handy while solving trigonometric functions because of their in-built sin, cos, and tan functions. However, if you want to be a math wizard and exercise your brain, you can do this simple math without a calculator easily.

Before learning how to challenge yourself with interesting trigonometry exercises, let’s understand its basics.

There are three sub-branches of trigonometry:

Plane Trigonometry deals with the two-dimensional plane of a triangle. It states that when enough sides and angles of triangles are known, the remaining can be calculated easily using trigonometric functions.

The two important laws mathematicians use to find the unknown sides and angles of the triangle are the law of sines and the law of cosines. To maintain consistency, the angles of the triangles are named A, B, and C; whereas, the sides of the triangle are called a, b, and c.

Spherical Trigonometry involves the study of spherical triangles, formed due to the intersection of three big circle arcs on the surface of the sphere. Spherical triangles are different from planar ones because the sum of their angles exceeds the sum of angles in a flat triangle.

Functions of spherical triangles include the angles (A, B, and C) sides (a, b, and c), and the dihedral angles (α, β, and γ).

A list of common spherical trigonometry formulas is given below:

In a right-angled triangle, we have the following three sides:

**Perpendicular**— the vertical side beside the right angle i.e., a.

**Hypotenuse** — the longest side opposite to the right angle i.e., c.

**Base** — The adjacent side to the angle θ i.e., b.

Trigonometry introduces three primary functions – sine (sin), cosine (cos), and tangent (tan). These functions define the relationship between angles and the sides of a triangle.

**Sine (sin θ)**: In a right triangle with angle θ, the sine ratio is calculated as the ratio of the length of the side opposite the angle to the hypotenuse. Mathematically, it can be expressed as:

Sin (θ)= Opposite sideHypotenuse

**Cosine (cos θ)**: For the same right triangle, the cosine ratio is the ratio of the length of the adjacent side to the hypotenuse. The formula is:

Cos (θ)= Adjacent sideHypotenuse

**Tangent (tan θ)**: Tangent is the ratio of the length of the side opposite the angle to the length of the adjacent side. The formula is:

Tan (θ)= Opposite sideAdjacent

**Tip to Remember: SOH CAH TOA**

Here, S stands for sine, C for cosine, and T for the tangent. You can remember this simple trick to recall the above formulas easily.

Angles are measured in degrees and radians. It's like deciding how big a slice of pizza you want from a whole pizza (circle).

**Degrees**: One full circle is 360 degrees. So, an angle can be measured as θ degrees.

**Radians**: Radians are a different way to measure angles. One full circle is 2π radians. The conversion between degrees (°) and radians (rad) is given by:

Radians = (180) x Degrees

Trigonometric tables are like cheat sheets for trigonometry. They're pre-calculated tables of values for sine, cosine, and tangent functions at specific angles. These tables were incredibly handy in the days before calculators and provided quick access to trigonometric values for various angles.

Using these tables involves finding the angle you're interested in. Here's how it works:

**Sine (sin)**: If you want to find the sine of an angle, locate the angle in the table, and read off the corresponding sine value.

**Cosine (cos)**: Similarly, for cosine, find the angle and read off the cosine value.

**Tangent (tan)**: To find the tangent, divide the sine value by the cosine value for the given angle.

For example, if you have an angle of 30 degrees, you'd find the row for 30 degrees in the sine column to get the sine value. Repeat this process for cosine and tangent.