Trigonometry Class 10: Formulas, Ratios & Solved Examples

Trigonometry class 10 is the study of the relationship between the sides and angles of a right-angled triangle. It covers two NCERT chapters: Introduction to Trigonometry and Some Applications of Trigonometry. Because the chapter is formula-based, a few hours of smart revision can fetch full marks. First, learn the six ratios. Next, memorise the table of standard angles. Finally, practise the identities and the heights-and-distances problems.

What Is Trigonometry? Basic Ideas for Class 10

The word comes from Greek and means “measuring triangles”. Here, we use only right-angled triangles. The side opposite the right angle is the hypotenuse. For a chosen angle θ, the side facing θ is the opposite side. The remaining side, touching θ, is the adjacent side.

Since similar triangles have equal side ratios, the ratio depends only on the angle. As a result, one table of values serves every triangle.

The Six Trigonometric Ratios

There are six ratios. However, you really need to remember only three, because the other three are their reciprocals.

Ratio	Formula	Reciprocal
sin θ	opposite / hypotenuse	cosec θ = hypotenuse / opposite
cos θ	adjacent / hypotenuse	sec θ = hypotenuse / adjacent
tan θ	opposite / adjacent	cot θ = adjacent / opposite

Two more links are useful. First, tan θ = sin θ / cos θ. Second, cot θ = cos θ / sin θ. A common memory aid is “SOH-CAH-TOA”: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.

Finding All Ratios from One Ratio

Suppose sin A = 3/5. Then opposite = 3 and hypotenuse = 5, so adjacent = 4 by the Pythagoras theorem. Therefore, cos A = 4/5 and tan A = 3/4. Likewise, cosec A = 5/3, sec A = 5/4 and cot A = 4/3. This method works for any given ratio.

Trigonometric Ratios of Standard Angles

Some angles appear in almost every question. Therefore, you should learn the following values by heart.

First, at 0°, sin = 0, cos = 1 and tan = 0. Next, at 30°, sin = 1/2, cos = √3/2 and tan = 1/√3. Then, at 45°, sin and cos are both 1/√2, while tan = 1. After that, at 60°, sin = √3/2, cos = 1/2 and tan = √3. Finally, at 90°, sin = 1 and cos = 0, but tan is not defined.

A trick helps here. For sine, the values follow the pattern √0/2, √1/2, √2/2, √3/2 and √4/2. For cosine, simply reverse the order. Meanwhile, tan equals sin divided by cos, so you can rebuild it quickly. Also, cosec, sec and cot are reciprocals, so cosec 90° = 1 and sec 0° = 1, but cot 0° is not defined.

Trigonometric Ratios of Complementary Angles

Two angles are complementary if their sum is 90°. In a right triangle, the two acute angles are always complementary. Consequently, the sine of one angle equals the cosine of the other.

First, sin (90° − θ) = cos θ, and cos (90° − θ) = sin θ. Next, tan (90° − θ) = cot θ, and cot (90° − θ) = tan θ. Finally, sec (90° − θ) = cosec θ, and cosec (90° − θ) = sec θ.

For example, sin 65° = cos 25°. Likewise, tan 40° × tan 50° = tan 40° × cot 40° = 1. Many board questions use this idea, so practise it often.

Trigonometric Identities

An identity is an equation that holds true for every value of the angle. The three identities below come from the Pythagoras theorem.

1. sin²θ + cos²θ = 1. Divide it by cos²θ, and you get the second one.
2. 1 + tan²θ = sec²θ. Divide the first by sin²θ, and you get the third.
3. 1 + cot²θ = cosec²θ.

How to Prove an Identity

Follow a fixed routine. First, start with the more complicated side. Next, convert everything into sine and cosine. Then simplify step by step until both sides match. Finally, write “Hence proved”. Never move terms across the equals sign, because that assumes the result you want to prove.

Solved Examples

Example 1. Find the value of sin 30° + cos 60° × tan 45°.
Solution: sin 30° = 1/2, cos 60° = 1/2 and tan 45° = 1. So the value = 1/2 + (1/2 × 1) = 1.

Example 2. Prove that (1 − sin²θ) × sec²θ = 1.
Solution: since 1 − sin²θ = cos²θ, the left side becomes cos²θ × sec²θ. Because sec θ = 1/cos θ, the product is 1. Hence proved.

Example 3. If tan θ = 1/√3, find sin θ and cos θ.
Solution: tan θ = 1/√3 means θ = 30°. Therefore, sin θ = 1/2 and cos θ = √3/2.

Some Applications of Trigonometry: Heights and Distances

The second chapter uses ratios to find heights and distances without direct measurement. Let us first learn the key terms.

• Line of sight: it is the line from the observer’s eye to the object.
• Angle of elevation: it is the angle above the horizontal when you look up at an object.
• Angle of depression: it is the angle below the horizontal when you look down at an object.

Steps to Solve Heights and Distances Problems

1. Draw a clear figure and mark the right angle.
2. Label the given angle, height and distance.
3. Choose the ratio that links the known and unknown sides.
4. Solve the equation and write the unit.

Example 4. A tower casts a shadow 20 m long when the sun’s elevation is 45°. Find the height.
Solution: tan 45° = height / 20. Since tan 45° = 1, the height is 20 m.

Example 5. From a point 30 m from the foot of a tower, the angle of elevation is 60°. Find the height. (Use √3 = 1.73.)
Solution: tan 60° = h / 30, so h = 30√3 = 51.9 m approximately.

Common Mistakes in Trigonometry Class 10

• Students confuse the opposite and adjacent sides when the angle changes.
• Many write sin²θ as sin θ², but sin²θ means (sin θ)².
• Some forget that the angle of depression is measured from the horizontal, not from the vertical.
• Because tan 90° is not defined, never substitute it blindly into an expression.
• When a figure is missing, marks are often lost even if the algebra is right.

How to Revise Trigonometry Class 10 in Three Days

On day one, learn the ratio table, the standard angles and the three identities. Next, on day two, solve all NCERT examples and exercise questions on values and complementary angles. Finally, on day three, practise two proofs and four heights-and-distances sums with neat figures. If you want more chapters, browse the Class 10 Maths notes or the wider Class 10 notes hub. When your study plan feels heavy, read how to stop procrastinating.

Wrap-Up

Trigonometry class 10 rewards practice more than memory. Learn the table and the identities, draw a figure for every word problem, and check each answer for units. Then move on to the next topic from the Class 10 Maths category, or go back to the The Note homepage.