Trigonometric Ratios Explained: Formulas, Identities & Practice

What is Trigonometry?

Trigonometric ratios are right-angled triangle functions that show the relationship between angles and sides.

Topics Trigonometry Covers

The following are the topics trigonometry covers:

• Right Triangle Trigonometry
• Unit Circle
• Graphs of Trig Functions
• Trigonometric Identities
• Inverse Trig Functions
• Solving Triangles
• Trigonometric Equations
• Real World Applications

What Skills Does Trigonometry Develop?

Trigonometry in Mathematics develops core skills like problem-solving abilities, logical and analytical thinking, a strong foundation of advanced mathematics, critical thinking, and spatial reasoning.

What are Trigonometric Ratios?

If we have a right-angle triangle, the length of each side of the triangle tells how steep or wide the given angle is, and trigonometric ratios are the simple comparison of those side lengths of the angle.

The basic trigonometric ratios include sine, cosine, and tangent.

However, other trigonometric ratios include cosec, sec, and cot.

How to Find Trigonometric Ratios?

In a right-angle triangle (an angle = 90°), find out

• Opposite: The side opposite to θ
• Adjacent: The side next to θ
• Hypotenuse: The side always opposite the right angle

Identifying adjacent, opposite and hypotenuse will help you find trigonometric ratios.

Trigonometric Ratio Formulas

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

For Example

In the above given right-angle triangle

c = Hypotenuse = 5

a = Opposite = 3

b = Adjacent = 4

The trigonometric ratios for the above right-angle triangle will be

sin (37°) = opposite/hypotenuse = 3/5

cos (37°) = adjacent/hypotenuse = 4/5

tan θ = opposite/adjacent = 3/4

SOH-CAH-TOA Technique

SOH-CAH-TOA is a simple way to learn how to memorize trigonometric ratios without confusion.

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

How to Find Trigonometric Ratios?

Besides the core three sine, cosine and tangent, other trigonometric ratios include Cotangent, Secant, and Cosecant.

As we know how to find hypotenuses, perpendicular or opposite (same), and adjacent, it will be easy to find all trig ratios.

The following are the formulas to find other trigonometric ratios.

• Sine = sin θ = opposite/hypotenuse
• Cosine = cos θ = adjacent/hypotenuse
• Tangent = tan θ = opposite/adjacent
• Cosecant = cscθ = Hypothesis/Opposite
• Secant = secθ =Hypothesis/Adjacent
• Cotangent = cot θ = Adjacent/Opposite

For Example

In Figure 1.2, the given sides are

Hypotenuse = 5

Opposite = 3

Adjacent = 4

So the trigonometry ratios will be

sin θ = opposite/hypotenuse = 3/5

cos θ = adjacent/hypotenuse = 4/5

tan θ = opposite/adjacent = 3/4

cscθ = Hypothesis/Opposite = 5/3

secθ = Hypothesis/Adjacent = 5/4

cot θ = Adjacent/Opposite = 4/3

All Trigonometric Ratios Formulas

Below is a cleaner look at all trigonometric ratios with their formulas. Students can practice using a trigonometric ratios formula worksheet to strengthen their understanding.

Trigonometric Ratios Table

Below is the trigonometric ratios table with respect to angles 0°, 30°, 45°, 60°, and 90°.

Trigonometric Identities

Trigonometric identities are true for every value of the variable recurring on both sides of the expression or equation. Trigonometric identities involve all trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent.
Trigonometric identities include:

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

Difference Between Trig Ratios and Trig Identities

Fundamental Trigonometric Identities

Below are common trigonometric identities used to solve complex maths problems.

Reciprocal Identities

• sin θ = 1 / cosec θ
• cosec θ = 1 / sin θ
• cos θ = 1 / sec θ
• sec θ = 1 / cos θ
• tan θ = 1 / cot θ
• cot θ = 1 / tan θ

Quotient Identities

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

Pythagorean Trig Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Double-Angle Trigonometric Identities

Sine

• sin 2θ = 2 sin θ cos θ

Cosine

• cos 2θ = cos²θ − sin²θ
• cos 2θ = 1 − 2 sin²θ
• cos 2θ = 2 cos²θ − 1

Tangent

• tan 2θ = 2 tan θ / (1 − tan²θ)

Cotangent

• cot 2θ = (cot²θ − 1) / (2 cot θ)

Half-Angle Trigonometric Identities

Sine

• sin(θ/2) = ± √[(1 − cos θ) / 2]

Cosine

• cos(θ/2) = ± √[(1 + cos θ) / 2]

Tangent

• tan(θ/2) = ± √[(1 − cos θ) / (1 + cos θ)]
• tan(θ/2) = (1 − cos θ) / sin θ
• tan(θ/2) = sin θ / (1 + cos θ)

Cotangent

• cot(θ/2) = ± √[(1 + cos θ) / (1 − cos θ)]
• cot(θ/2) = (1 + cos θ) / sin θ
• cot(θ/2) = sin θ / (1 − cos θ)

Sum and Difference Trigonometric Identities

Sine

• sin(A + B) = sin A cos B + cos A sin B
• sin(A − B) = sin A cos B − cos A sin B

Cosine

• cos(A + B) = cos A cos B − sin A sin B
• cos(A − B) = cos A cos B + sin A sin B

Tangent

• tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
• tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

Product-to-Sum Trigonometric Identities

Sine × Sine

• sin A sin B = ½ [cos(A − B) − cos(A + B)]

Cosine × Cosine

• cos A cos B = ½ [cos(A − B) + cos(A + B)]

Sine × Cosine

• sin A cos B = ½ [sin(A + B) + sin(A − B)]
• cos A sin B = ½ [sin(A + B) − sin(A − B)]

Sum-to-Product Identities

Sine

• sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)
• sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)

Cosine

• cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)
• cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

Triple-Angle Identities

Sine

• sin 3θ = 3 sin θ − 4 sin³θ

Cosine

• cos 3θ = 4 cos³θ − 3 cos θ

Tangent

• tan 3θ = (3 tan θ − tan³θ) / (1 − 3 tan²θ)

Power-Reduction (Reduction) Identities

• sin²θ = (1 − cos 2θ) / 2
• cos²θ = (1 + cos 2θ) / 2
• tan²θ = (1 − cos 2θ) / (1 + cos 2θ)

Co-Function Identities

• sin(90° − θ) = cos θ
• cos(90° − θ) = sin θ
• tan(90° − θ) = cot θ
• sec(90° − θ) = cosec θ
• cosec(90° − θ) = sec θ
• cot(90° − θ) = tan θ

Negative-Angle Identities

• sin(−θ) = −sin θ
• cos(−θ) = cos θ
• tan(−θ) = −tan θ

Periodic Identities

• sin(θ + 2π) = sin θ
• cos(θ + 2π) = cos θ
• tan(θ + π) = tan θ

Trigonometry Solved Questions Samples

Question 1: Find the value

If sin θ = 3/5, where θ is acute, find cos θ and tan θ.

Solution:

Given:

sin θ = Opposite / Hypotenuse = 3/5

So:

Opposite = 3

Hypotenuse = 5

Using Pythagoras:

Adjacent = √(5² − 3²)

Adjacent = √(25 − 9) = √16 = 4

Now:

• cos θ = Adjacent / Hypotenuse = 4/5
• tan θ = Opposite / Adjacent = 3/4

Answer:

cos θ = 4/5

tan θ = 3/4

Question 2: Evaluate

sin 30° + cos 60°

Solution:

We know:

sin 30° = 1/2

cos 60° = 1/2

So:

sin 30° + cos 60° = 1/2 + 1/2 = 1

Answer: 1

Question 3: Prove that

(1 − cos 2θ) / sin 2θ = tan θ

Solution:

(1 − cos 2θ) / sin 2θ

= (2 sin²θ) / (2 sin θ cos θ)

= sin θ / cos θ

= tan θ

Hence Proved

Question 4: Evaluate

sin θ / (1 + cos θ) + (1 + cos θ) / sin θ

Solution:

sin θ / (1 + cos θ) + (1 + cos θ) / sin θ

= sin²θ / (sin θ(1 + cos θ)) + (1 + cos θ)² / (sin θ(1 + cos θ))

= (sin²θ + (1 + cos θ)²) / (sin θ(1 + cos θ))

= (sin²θ + (1 + 2 cos θ + cos²θ)) / (sin θ(1 + cos θ))

= ((sin²θ + cos²θ) + 1 + 2 cos θ) / (sin θ(1 + cos θ))

= (1 + 1 + 2 cos θ) / (sin θ(1 + cos θ))

= 2(1 + cos θ) / (sin θ(1 + cos θ))

= 2 / sin θ

= 2 cosec θ

Question 4: Evaluate

Simplified Identity

Find the value of 9 csc^2θ − 9 cot^2θ.

Solution:

Factor out 9:

9(csc^2θ − cot^2θ)

Using the identity:

1 + cot^2θ = csc^2θ

So,

csc^2θ − cot^2θ = 1

Therefore:

9(1) = 9

Trigonometry Sample MCQs

1. If sin θ = 3/5, where θ is acute, what is cos θ?
   
   
   
   • A) 3/4
   
   
   • B) 4/5
   
   
   • C) 5/3
   
   
   • D) 5/4
   
   
   
   

   Correct Answer: B) 4/5

   
   
2. Evaluate: sin 30° + cos 60°
   
   
   
   • A) 0
   
   
   • B) 1/2
   
   
   • C) 1
   
   
   • D) 2
   
   
   
   

   Correct Answer: C) 1

   
   
3. Which of the following is equal to 1 + tan²θ?
   
   
   
   • A) cosec²θ
   
   
   • B) sec²θ
   
   
   • C) cot²θ
   
   
   • D) sin²θ
   
   
   
   

   Correct Answer: B) sec²θ

   
   
4. Find the value of: (sec θ − tan θ)(sec θ + tan θ)
   
   
   
   • A) 0
   
   
   • B) 1
   
   
   • C) sec²θ
   
   
   • D) tan²θ
   
   
   
   

   Correct Answer: B) 1

   
   
5. Evaluate: sin 2θ / (1 + cos 2θ)
   
   
   
   • A) sin θ
   
   
   • B) cos θ
   
   
   • C) tan θ
   
   
   • D) cot θ
   
   
   
   

   Correct Answer: C) tan θ

   
   

How Does Mixt Academy Help Students Excel in Mathematics?

Mixt Academy is providing expert tutors for all subjects, including mathematics. We will note the student’s academic requirement first and then match each student with the most suitable online maths tutors who are connected globally on our platform. Flexible timings and easy-to-understand lectures help students achieve higher grades in tests or exams.
So, either you need help with your daily homework, or it’s the IGCSE, GCSE or A level, we provide customised support for all curricula and subjects.

Mastering Trigonometry Made Easy

Trigonometry may seem challenging at the beginning, but with practice and the right steps, it becomes much easier to understand.

By learning trigonometric ratios, formulas, and identities, students can solve many math problems quickly and correctly.

Remember to practice regularly, use shortcuts like SOH-CAH-TOA, and understand each step instead of memorising blindly.

With time and practice, mastering trigonometric ratios will feel simple and rewarding.