What this chapter is about
Trigonometry means "measurement of triangles." This chapter introduces you to a branch of mathematics that connects angles with ratios of sides in a right-angled triangle. You will learn how the size of an acute angle in a right triangle determines fixed ratios between pairs of sides, no matter how large or small the triangle is.
At Class 10, you are ready for this because you already understand similar triangles and the Pythagoras theorem. Trigonometry builds directly on these ideas. The ratios you learn here — sine, cosine, tangent and their reciprocals — appear throughout higher mathematics, physics, engineering and navigation.
After studying this chapter, you should be able to define the six trigonometric ratios, calculate them for any acute angle in a right triangle when side lengths are known, use the standard values for 0°, 30°, 45°, 60° and 90°, and apply fundamental identities to simplify expressions or prove results.
Key ideas
- In a right-angled triangle, for a given acute angle, the ratio of any two sides stays the same regardless of the triangle's size; this constant ratio is called a trigonometric ratio.
- The three primary ratios are: sin θ = (side opposite to θ) / hypotenuse, cos θ = (side adjacent to θ) / hypotenuse, tan θ = (side opposite) / (side adjacent).
- The three reciprocal ratios are: cosec θ = 1 / sin θ, sec θ = 1 / cos θ, cot θ = 1 / tan θ.
- Standard angle values must be memorised: for example, sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 90° = 1, cos 0° = 1.
- Complementary-angle relations: sin (90° − θ) = cos θ, cos (90° − θ) = sin θ, tan (90° − θ) = cot θ, and similarly for the reciprocal ratios.
- Three fundamental identities hold for every acute angle θ: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
Formulas and facts to remember
- Formula: sin θ = opposite / hypotenuse · Meaning: Ratio of the side facing the angle to the longest side.
- Formula: cos θ = adjacent / hypotenuse · Meaning: Ratio of the side next to the angle to the longest side.
- Formula: tan θ = opposite / adjacent = sin θ / cos θ · Meaning: Ratio of the side facing the angle to the side next to it.
- Formula: cosec θ = 1 / sin θ · Meaning: Reciprocal of sine.
- Formula: sec θ = 1 / cos θ · Meaning: Reciprocal of cosine.
- Formula: cot θ = 1 / tan θ = cos θ / sin θ · Meaning: Reciprocal of tangent.
- Formula: sin²θ + cos²θ = 1 · Meaning: Pythagorean identity linking sine and cosine.
- Formula: 1 + tan²θ = sec²θ · Meaning: Derived from the first identity by dividing by cos²θ.
- Formula: 1 + cot²θ = cosec²θ · Meaning: Derived from the first identity by dividing by sin²θ.
- Formula: sin (90° − θ) = cos θ, cos (90° − θ) = sin θ · Meaning: Complementary-angle relations.
Standard values:
- θ: sin θ · 0°: 0 · 30°: 1/2 · 45°: 1/√2 · 60°: √3/2 · 90°: 1
- θ: cos θ · 0°: 1 · 30°: √3/2 · 45°: 1/√2 · 60°: 1/2 · 90°: 0
- θ: tan θ · 0°: 0 · 30°: 1/√3 · 45°: 1 · 60°: √3 · 90°: not defined
Worked examples
Example 1. In a right-angled triangle PQR, angle Q = 90°, PQ = 5 cm and QR = 12 cm. Find sin R, cos R and tan P.
Step 1: Identify the hypotenuse. Using Pythagoras: PR = √(5² + 12²) = √(25 + 144) = √169 = 13 cm.
Step 2: For angle R, the opposite side is PQ = 5 cm and the adjacent side is QR = 12 cm. sin R = 5/13, cos R = 12/13.
Step 3: For angle P, the opposite side is QR = 12 cm and the adjacent side is PQ = 5 cm. tan P = 12/5.
Example 2. Evaluate without tables: sin 60° cos 30° + cos 60° sin 30°.
Step 1: Substitute standard values. sin 60° = √3/2, cos 30° = √3/2, cos 60° = 1/2, sin 30° = 1/2.
Step 2: Compute each product. (√3/2) × (√3/2) = 3/4. (1/2) × (1/2) = 1/4.
Step 3: Add: 3/4 + 1/4 = 1.
Example 3. Prove that (1 − cos²A)(1 + cot²A) = 1.
Step 1: Replace 1 − cos²A using identity: sin²A + cos²A = 1 gives 1 − cos²A = sin²A.
Step 2: Replace 1 + cot²A using identity: 1 + cot²A = cosec²A.
Step 3: The expression becomes sin²A × cosec²A = sin²A × (1/sin²A) = 1.
Hence proved.
Common mistakes
Confusing opposite and adjacent sides → always label sides with respect to the angle you are working with, not the right angle.
Writing sin 60° = 60 or treating sin as a multiplier → sin is a ratio, not a quantity to multiply by the angle.
Using tan 90° as a finite number → tan 90° is undefined because cos 90° = 0.
Forgetting to square correctly in identities: writing sin θ + cos θ = 1 instead of sin²θ + cos²θ = 1 → the identity involves squares.
Mixing up complementary relations: writing sin (90° − θ) = sin θ → it equals cos θ, not sin θ.
Quick revision
- sin²θ + cos²θ = 1 is the master identity; the other two follow from it.
- tan θ = sin θ / cos θ; cot θ = cos θ / sin θ.
- Memorise standard values for 0°, 30°, 45°, 60°, 90° — most problems use them.
- Complementary angles add to 90°; their sine and cosine swap.
- Always identify which side is opposite and which is adjacent before writing a ratio.