Solve Trigonometry Functions Without a Calculator

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.

Branches of Trigonometry

There are three sub-branches of trigonometry:

1. Plane 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.

2. Spherical Trigonometry

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:

Basic Trigonometric Concepts

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.

1. Trigonometric Functions

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.

2. Angles and Measurements

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

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.

Tips for Memorizing Common Angles:

While tables are convenient, memorizing values for common angles can save time. Here are some tips:

• 0°, 30°, 45°, 60°, 90°: These are frequently used angles. Memorize their sine and cosine values.
• Use Patterns: Notice patterns in the values. For example, the sine of 30° is always half the sine of 60°.
• Practice: Regularly practice and use trigonometric functions to reinforce memorization.

Special Triangles

Special right triangles have angles that result in simple, easily memorable ratios. The two main types are the 30-60-90 triangle and the 45-45-90 triangle.

30-60-90 Triangle: In this triangle, the angles are 30°, 60°, and 90°. The sides are in the ratio 1: √3: 2. 

To find it, draw a right triangle with two additional angles that are roughly 30 and 60 degrees each to create a 30-60-90 triangle. You already know the ratios of sides. The smallest angle (30 degrees) is opposite the smallest side (QK). The hypotenuse, which has the largest side (PK), is located in opposition to the largest angle (90 degrees). The remaining 60-degree angle is opposite the square root of 3 (PQ).

45-45-90 Triangle: This triangle has angles of 45°, 45°, and 90°. The sides are in the ratio 1: 1: √2.

Draw a right triangle with two equal angles in the 45-45-90 triangle. The length of the hypotenuse is the square root of two; the remaining two sides are one. In order to calculate the cosine of 60 degrees, one would draw the triangle 30-60-90 and observe that the hypotenuse is two and the adjacent side is one. Therefore, 1/2 represents the cosine of 60 degrees.

Relevant Read: Choose The Right Math Tutor For Your Child

Importance of Understanding Trigonometric Functions Without a Calculator

A calculator has solved most of our problems but that does not mean that manual calculations have no benefits. You cannot always rely on using a calculator for everyday calculations. Sometimes, you have to do quick maths in certain situations where no calculator is available. 

. Let’s have a look at some of them.

• Manual trigonometry is a tool that promotes critical thinking. By understanding the ideas, people are able to see issues and come up with creative solutions rather than depending just on a calculator.
• Beyond using a calculator, trigonometry enhances problem-solving skills. It encourages people to break down complex problems, formulate strategies, and apply mathematical principles. 
• These manual calculations help in advanced studies and have diverse applications in various fields.

Where is Trigonometry used?

Trigonometry is used in various fields and by different experts. Some of them are given below:

1. Navigation and Astronomy
2. Surveying and Construction
3. Engineering and Physics
4. Computer Graphics
5. Architecture

Interesting Information: If you want to apply a trigonometric function in a real-life scenario then take an example of a tree. Can you calculate the height of a tree without having to measure it from any measuring device?

Yes!

If you know the distance between you and the tree and the angle where you are looking at the tree from the ground, then you can easily calculate the height of the tree.

Here’s how you can do it:

You know the formula of the tan function above. 

Tan (θ)= HeightDistance between you and the tree

Distance between you and the tree= HeightTan (θ)

Let’s assume the distance is 50 m and the angle is 45 degrees. The Height of the tree will be:

Height = Distance between you and the tree x Tan (θ)

Height = 50 x tan 45

Height = 50 m (since tan 45 = 1)

Choose a Tutor That Sharpens Your Skills

When practising trigonometry, the right tutor can make all the difference.

Meet My Tutor Source (MTS), a private tutoring company dedicated to sharpening your trigonometric skills. We provide one-on-one private tutoring sessions with experienced tutors all over the world. The flexibility of schedules and online options provided by MTS makes learning convenient, fitting seamlessly into your busy life.

So, if you want to sharpen your skills and be a manual trigonometry wizard, choose a math tutor from MTS. Book your lesson today with an expert.

Final Words

We've covered the basics of dealing with the sides and angles of triangles, explored useful tricks like using tables for quick solutions, and highlighted the importance of understanding trigonometry beyond just numbers.

The key takeaway is not about making math fancier; it's about making it more useful in everyday situations. By doing manual trigonometry, you're not just cramming numbers – you're sharpening your brain for problem-solving in fields like navigation, construction, computer graphics, and more. 

So, the next time you opt for manual calculations, remember it's a straightforward way to boost your practical math skills. For this, you can always take a free private class with an expert!