Trigonometric Functions in Algebra and Geometry

The concept of trigonometric functions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From calculating heights and distances in geometry to solving periodic problems in science and engineering, trigonometric functions are essential tools for students from Class 9, 10, 11, and onwards. This topic is especially important for board exams and competitive tests like JEE or NEET.

What Is Trigonometric Functions?

A trigonometric function is defined as a mathematical function that relates an angle of a right triangle to the ratios of its sides. You’ll find this concept applied in areas such as geometry, physics (like understanding waves and oscillations), and engineering. The six main trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each function describes a different relationship using the sides of a right-angled triangle.

Key Formula for Trigonometric Functions

Here’s the standard formulae for the six trigonometric functions for an angle θ in a right-angled triangle:

Function	Formula
sin θ	Opposite / Hypotenuse
cos θ	Adjacent / Hypotenuse
tan θ	Opposite / Adjacent
cosec θ	Hypotenuse / Opposite
sec θ	Hypotenuse / Adjacent
cot θ	Adjacent / Opposite

Some of the most commonly used trigonometric identities include:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

Standard Values of Trigonometric Functions

Degrees	Radians	sin θ	cos θ	tan θ	cosec θ	sec θ	cot θ
0°	0	0	1	0	Undefined	1	Undefined
30°	π/6	1/2	√3/2	1/√3	2	2/√3	√3
45°	π/4	1/√2	1/√2	1	√2	√2	1
60°	π/3	√3/2	1/2	√3	2/√3	2	1/√3
90°	π/2	1	0	Undefined	1	Undefined	0

Cross-Disciplinary Usage

Trigonometric functions are not only useful in Mathematics, but also play an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see trigonometry used in projectile motion, alternating current, and digital signal processing questions.

Step-by-Step Illustration

Let’s solve an example using the tangent function:

1. A boy is standing 10 metres from a tree. The angle of elevation to a bird on top is 20°.

2. Let x be the height from ground to the bird.

3. Formula: tan θ = Opposite / Adjacent

4. Substitute the values: tan(20°) = x / 10

5. x = 10 × tan(20°) ≈ 10 × 0.364 = 3.64 meters

6. Final Answer: The bird is sitting approx. 3.6 m from the ground.

Speed Trick or Vedic Shortcut

Here’s a quick way to remember the three main trigonometric ratios: SOH-CAH-TOA.

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

This trick is used by students for fast recall in competitive and board exams.

Try These Yourself

• Find sin 30°, sin 45°, sin 60° using the value table above.
• If tan θ = 1, what is the value of θ between 0° and 90°?
• Is sec 90° defined?
• Write the cofunction identity for cos(90° – θ).

Frequent Errors and Misunderstandings

• Confusing the numerator and denominator in the main trigonometric ratios.
• Not knowing that tan 90° (and similar values) can be undefined.
• Mixing degrees and radians when using calculators.

Relation to Other Concepts

The idea of trigonometric functions connects closely with topics such as the unit circle and trigonometric identities. Mastering this helps with understanding graphs, calculus (like differentiation and integration of trig functions), and many advanced chapters in school Maths and competitive exams. For further study, visit Trigonometric Ratios and Trigonometric Equations.

Classroom Tip

A quick way to remember trigonometric functions is with the SOH-CAH-TOA rhyme—and by repeatedly practicing value tables for common angles (like 0°, 30°, 45°, and 60°). Vedantu’s teachers often use colored charts and fun mnemonics to simplify learning during live classes.

We explored trigonometric functions—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept. For more, check related topics like Applications of Trigonometry, Trigonometry Overview, and Trigonometry Table.