Trigonometry Formulas Explained with Examples

Trigonometry formulas look like a long list to memorise, but they grow from just a few ideas. If you understand where each one comes from, you can rebuild it during an exam or on a site visit instead of hoping memory holds. This article groups the key formulas, explains what they mean in plain words, and gives worked examples with numbers that suit Indian classrooms, entrance exams and building work.

Group 1: The Basic Ratios

For an angle theta in a right triangle with perpendicular P, base B and hypotenuse H:

sin theta = P/H, cos theta = B/H, tan theta = P/B

cosec theta = H/P, sec theta = H/B, cot theta = B/P

Also, tan theta = sin theta / cos theta and cot theta = cos theta / sin theta. These links save time because if you know sin and cos, you automatically know the other four.

Group 2: The Pythagorean Identities

From the Pythagoras theorem (P squared + B squared = H squared), dividing through by H squared gives the most important identity:

sin squared theta + cos squared theta = 1

Dividing by cos squared gives 1 + tan squared theta = sec squared theta, and dividing by sin squared gives 1 + cot squared theta = cosec squared theta. These three identities appear in nearly every CBSE Class 10 and Class 11 paper.

Expert insight: When stuck on a proof, rewrite everything in sin and cos. Many identity questions collapse into simple algebra after that step.

Example: Find cos theta when sin theta = 3/5

cos squared theta = 1 – (3/5) squared = 1 – 9/25 = 16/25, so cos theta = 4/5 (taking the positive value for an acute angle). Then tan theta = (3/5) / (4/5) = 3/4. This is the familiar 3-4-5 triangle.

Group 3: Complementary Angle Relations

In a right triangle the two acute angles add to 90 degrees, so:

sin(90 - theta) = cos theta, tan(90 - theta) = cot theta, sec(90 - theta) = cosec theta

For instance, sin 70 degrees equals cos 20 degrees. A typical exam question asks for sin 35 / cos 55, which equals 1 because cos 55 = sin 35.

Group 4: Sum and Difference Formulas

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

cos(A + B) = cos A cos B - sin A sin B

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Replace the plus with a minus and flip the matching signs for A – B. These are studied in Class 11 and are heavily used in JEE problems.

Example: Find sin 75 degrees

Write 75 as 45 + 30. sin 75 = sin 45 cos 30 + cos 45 sin 30 = (0.7071 x 0.8660) + (0.7071 x 0.5) = 0.6124 + 0.3536 = 0.9659. The calculator value matches.

Group 5: Double Angle Formulas

sin 2A = 2 sin A cos A

cos 2A = cos squared A - sin squared A = 2 cos squared A - 1 = 1 - 2 sin squared A

tan 2A = 2 tan A / (1 - tan squared A)

Example: if sin A = 3/5 and cos A = 4/5, then sin 2A = 2 x 3/5 x 4/5 = 24/25 = 0.96, and cos 2A = 16/25 – 9/25 = 7/25 = 0.28.

Group 6: Rules for Any Triangle

When a triangle has no right angle, use these with sides a, b, c opposite angles A, B, C:

Sine rule: a / sin A = b / sin B = c / sin C

Cosine rule: a squared = b squared + c squared - 2bc cos A

Area = (1/2) x b x c x sin A

Example: Land plot area

A triangular plot has two boundary sides of 30 m and 45 m with an angle of 50 degrees between them. Area = 0.5 x 30 x 45 x sin 50 = 0.5 x 1350 x 0.7660 = 517 square metres. In many Indian states, plots are also quoted in square feet or gaj, so you may convert 517 sq m into about 5,565 sq ft.

Table of Key Formulas

Purpose	Formula
Basic identity	sin squared + cos squared = 1
Tan from sin and cos	tan = sin / cos
Complement	sin(90 – x) = cos x
Double angle	sin 2A = 2 sin A cos A
Sine rule	a / sin A = b / sin B
Cosine rule	a squared = b squared + c squared – 2bc cos A
Degrees to radians	radians = degrees x pi / 180

Radians and Degrees

One full turn is 360 degrees or 2 pi radians. So 180 degrees equals pi radians, and 90 degrees equals pi/2. In Class 11, trigonometric functions of real numbers use radians, and most calculus and physics problems also use radians. The degree calculator can help you convert quickly.

Applications You Will Actually See

Civil engineers use these formulas for slopes, roof pitches and road gradients. Architects use them for staircase design, where the rise and going must fit a comfortable angle. Electricians use sine waves to describe AC current, which in India alternates at 50 Hz. Surveyors use the sine and cosine rules in triangulation. Game and app developers use them for rotation and animation. In each case a calculator such as the trigonometry calculator handles the number work while you apply the formula.

Benefits of Understanding the Formulas

When you know how formulas connect, you need to remember far fewer of them. A single identity can generate several others, and complement relations halve the table of values you must learn. This also reduces panic in exams. You can verify results by substituting sample angles, such as 30 or 45 degrees, into both sides of an identity.

Challenges and Limitations

Sign errors in sum and difference formulas, quadrant mistakes for angles above 90 degrees and confusion between degrees and radians cause many lost marks. Some formulas, such as the double angle for tan, break down when the denominator is zero. Always check domains, and remember that an inverse function returns only one of several possible angles.

Common Mistakes

1. Writing sin(A + B) = sin A + sin B. This is false; use the full sum formula.
2. Dropping the square in the identity: sin squared theta is not sin of theta squared.
3. Ignoring the sign in different quadrants. For angles beyond 90 degrees, use the ASTC (All, Sin, Tan, Cos) rule.
4. Mixing up the cosine rule signs, especially the minus before 2bc cos A.
5. Using the wrong angle mode in the calculator.

Best Practices

1. Derive when unsure. Start from sin squared plus cos squared equals 1.
2. Test with simple angles such as 30, 45 and 60 degrees.
3. Practise NCERT exercises and previous year board papers for pattern familiarity.
4. Keep a one-page formula sheet grouped as shown above.
5. Check final numbers with a calculator for accuracy.

Conclusion

Trigonometry formulas fall into six clear groups: ratios, identities, complements, sum and difference, double angle and triangle rules. Learn the logic behind each group, test them with easy angles, and use a calculator for the arithmetic. This approach reduces rote memory and builds the confidence needed for board exams, entrance tests and real measurement problems.

Which Quadrant Is the Angle In? (ASTC Rule)

For angles beyond 90 degrees, the sign of each ratio depends on the quadrant. In the first quadrant (0 to 90 degrees) All ratios are positive. In the second (90 to 180) only Sine and its reciprocal are positive. In the third (180 to 270) only Tangent and cotangent are positive. In the fourth (270 to 360) only Cosine and secant are positive. Students remember this as ASTC, or the phrase All Silver Tea Cups. Example: sin 150 degrees equals sin 30 degrees = 0.5 because 150 = 180 – 30 and sine is positive in the second quadrant, whereas cos 150 degrees equals negative cos 30 degrees = -0.866.

FAQs

What is the most important trigonometric identity?
It is sin squared theta plus cos squared theta equals 1. Other identities, such as 1 + tan squared theta = sec squared theta, come from dividing this one by cos squared or sin squared.

How do I find sin 75 degrees without a calculator?
Write 75 as 45 plus 30 and use the sum formula. sin 75 equals sin 45 cos 30 plus cos 45 sin 30, which is about 0.9659.

What is the cosine rule used for?
It finds a side when two sides and the included angle are known, or an angle when all three sides are known. The formula is a squared equals b squared plus c squared minus 2bc cos A.

How do I convert degrees to radians?
Multiply the degrees by pi and divide by 180. For example, 60 degrees is pi/3 radians, which is about 1.047.

Which trigonometry formulas are in the CBSE syllabus?
Class 10 covers ratios, standard values, complementary angle relations and three basic identities. Class 11 adds sum and difference formulas, double angles and general solutions.