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CBSE · Class 10 · Mathematics

Introduction to Trigonometry 8.3: Trigonometric Ratios of Some Specific Angles

5 concepts · 40 practice questions with answers · Free to read

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Values at 0°, 30°, 45°, 60°, 90°

From daily life

Your geometry box has two set squares, 45°-45°-90° and 30°-60°-90°. Their fixed side ratios are exactly where these values come from.

Simple definition

The 45° triangle has sides 1 : 1 : √2 and the 30°-60° triangle has sides 1 : √3 : 2. Reading ratios from them gives the standard values, which you must know by heart. At 0° and 90° the values come from the limiting triangle.

11√245°45°√31230°60°

In the figure: Set-square sides: 1, 1, √2 and 1, √3, 2.

Live figure

Values at 0°, 30°, 45°, 60°, 90°

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Rule 1: Learn the sin row — The sin values follow a simple pattern.

  • Easy: sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1.
  • Medium: cos is the same list backwards: 1, √3/2, 1/√2, 1/2, 0.
  • Exam-type: tan = sin ÷ cos: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Rule 2: Undefined values — A ratio with zero in the denominator has no value.

  • Easy: tan 90° and sec 90° are undefined because cos 90° = 0.
  • Medium: cot 0° and cosec 0° are undefined because sin 0° = 0.
  • Exam-type: sin 0° = 0 and cos 0° = 1, so tan 0° = 0.
Remember

In the first quadrant sin rises from 0 to 1 and cos falls from 1 to 0.

Common mistake

Mixing sin 30° = 1/2 with cos 30° = √3/2; check which is bigger: at 30° the cos is bigger.

Memory clue

"Sin counts up √0, √1, √2, √3, √4 over 2; cos counts down."

Quick table

angle θ0°30°45°60°90°
sin01/21/√2√3/21
cos1√3/21/√21/20
tan01/√31√3not defined

Practice questions

Easy
  1. 1What is the value of cos 30°?
  2. 2What is the value of cosec 90°?
  3. 3What is the value of sec 30°?
Medium
  1. 4What is the value of cosec 30°?
  2. 5What is the value of cot 60°?
  3. 6What is the value of tan 45°?
Challenging
  1. 7What is the value of cosec 60°?
  2. 8What is the value of tan 0°?
Show answers (try them yourself first)
  1. 1√3/2
  2. 21
  3. 32/√3
  4. 42
  5. 51/√3
  6. 61
  7. 72/√3
  8. 80

Evaluating simple expressions

From daily life

A shopkeeper converts every price to the same unit before adding. Here every ratio is first replaced by its number.

Simple definition

To evaluate an expression, replace each ratio by its standard value, then simplify with fractions and square roots as in ordinary arithmetic. Keep answers exact (like √3/2) unless a decimal is asked.

put valuessin 30° = 1/2simplifyfractionsanswer

In the figure: sin 30° + cos 60° = 1/2 + 1/2 = 1.

Rule 1: Add and subtract — Replace, then combine like fractions.

  • Easy: sin 30° + cos 60° = 1/2 + 1/2 = 1.
  • Medium: 2 tan 45° − cos 0° = 2 × 1 − 1 = 1.
  • Exam-type: tan 60° − 2 sin 60° = √3 − √3 = 0.

Rule 2: Products — Multiply the numbers, then add the products.

  • Easy: sin 60° cos 30° = (√3/2)(√3/2) = 3/4.
  • Medium: sin 30° cos 60° = (1/2)(1/2) = 1/4.
  • Exam-type: Sum of the last two: 3/4 + 1/4 = 1.
Remember

Write each standard value in brackets before multiplying, to avoid sign and fraction slips.

Common mistake

Writing sin 30° + cos 60° as sin 90°; angles inside the names are never added.

Memory clue

"Replace first, calculate second."

Practice questions

Easy
  1. 1Find the value of sin 0° + cos 45°.
  2. 2Find the value of cos 0° − tan 30°.
  3. 3Find the value of tan 30° − cos 0°.
Medium
  1. 4Find the value of cos 60° + cos 0°.
  2. 5Find the value of sin 90° + 4 cos 60° + 2 cot 45°.
  3. 6Find the value of 3 cosec 60° − 2 cot 60°.
Challenging
  1. 7Find the value of 2 sec 45° + 3 cos 30°.
  2. 8Find the value of 5 tan 0° + cos 45° − 2 cot 45°.
Show answers (try them yourself first)
  1. 11/√2
  2. 2(3 − √3)/3
  3. 3(√3 − 3)/3
  4. 43/2
  5. 55
  6. 64√3/3
  7. 7(4√2 + 3√3)/2
  8. 8(√2 − 4)/2

Squares, products and quotients

From daily life

Areas use squares, so the square of a ratio appears whenever a problem mixes lengths and areas.

Simple definition

sin² A means (sin A)². Replace the ratio by its value, square the whole value, and then combine. Squares of the surd values become whole fractions: (√3/2)² = 3/4.

valuesin 45° = 1/√2square1/√2 × 1/√2result= 1/2

In the figure: sin² 45° = (1/√2)² = 1/2.

Rule 1: Square the value — Square the numerator and the denominator separately.

  • Easy: sin² 45° + cos² 45° = 1/2 + 1/2 = 1.
  • Medium: tan² 30° = (1/√3)² = 1/3.
  • Exam-type: sin² 30° + cos² 30° = 1/4 + 3/4 = 1.

Rule 2: Quotients and bigger expressions — Work term by term, then combine.

  • Easy: tan 60° ÷ sin 60° = √3 ÷ (√3/2) = 2.
  • Medium: (sin 30° + cos 30°)² = 1/4 + 2(1/2)(√3/2) + 3/4 = 1 + √3/2.
  • Exam-type: 4 cot² 45° − sec² 60° − sin² 60° = 4 − 4 − 3/4 = −3/4.
Remember

sin² 30° + cos² 30° = 1 for every angle, not only 30°; this is the first identity.

Common mistake

Squaring only the top: (√3/2)² is 3/4, not √3/4.

Memory clue

"Square the whole fraction: top and bottom both."

Practice questions

Easy
  1. 1Find the value of tan² 0° + cos² 30°.
  2. 2Find the value of cos² 45° + tan² 60°.
  3. 3Find the value of cos² 30° + sin² 60°.
Medium
  1. 4Find the value of cos² 45° − cos² 30°.
  2. 5Find the value of 3 sec² 60° − sin 90°.
  3. 6Find the value of 4 cot 45° sin 30° + 2 cot 30° sin 90°.
Challenging
  1. 7Find the value of (cot 60° − cot 30°)/(sin 45° + cos 45°).
  2. 8Find the value of (sin 60° − cos 45°)/(cos 0° + cot 90°).
Show answers (try them yourself first)
  1. 13/4
  2. 27/2
  3. 33/2
  4. 4−1/4
  5. 511
  6. 62 + 2√3
  7. 7−√6/3
  8. 8(√3 − √2)/2

Finding the angle from a ratio or equation

From daily life

Like finding a house from its address, a standard value tells you which standard angle you are looking for.

Simple definition

Simplify until one ratio equals a standard value, then read the angle from the table. If the angle appears as 2A or 3A, solve for that and then divide. For a pair of equations, add and subtract the angles.

sin A = 1/2A = 30°cos A = 1/√2A = 45°tan A = √3A = 60°

In the figure: sin A = 1/2 gives A = 30°; cos A = 1/√2 gives A = 45°; tan A = √3 gives A = 60°.

Rule 1: Read the angle — Make one ratio equal a table value.

  • Easy: 2 sin A = 1 gives sin A = 1/2, so A = 30°.
  • Medium: sin A = cos A for an acute angle gives A = 45°.
  • Exam-type: tan 2A = √3 gives 2A = 60°, so A = 30°.

Rule 2: Two conditions — Find both angles, then add and subtract.

  • Easy: 2 cos 3A = 1 gives cos 3A = 1/2, 3A = 60°, so A = 20°.
  • Medium: tan (A − B) = 1/√3 and tan (A + B) = √3 give A − B = 30°, A + B = 60°.
  • Exam-type: Adding: 2A = 90°, so A = 45° and B = 15°.
Remember

Check the answer by putting the angle back into the original equation.

Common mistake

Stopping at 3A = 60° and writing A = 60°; remember to divide by 3.

Memory clue

"Find the ratio, name the angle, then un-multiply."

Practice questions

Easy
  1. 1If sec θ = 1 for an acute angle θ, find θ.
  2. 2If cos θ = √3/2 for an acute angle θ, find θ.
  3. 3If tan θ = 0 for an acute angle θ, find θ.
Medium
  1. 4If 4 cot θ = 4√3 for an acute angle θ, find θ.
  2. 5If 4 cosec θ = 4√2 for an acute angle θ, find θ.
  3. 6If 2 sin θ = 1 for an acute angle θ, find θ.
Challenging
  1. 7If cot (A + B) = 1/√3 and tan (A − B) = 1/√3, where 0° ≤ B ≤ A ≤ 90°, find A (in degrees).
  2. 8If cot (A + B) = 0 and tan (A − B) = 1/√3, where 0° ≤ B ≤ A ≤ 90°, find A (in degrees).
Show answers (try them yourself first)
  1. 10°
  2. 230°
  3. 30°
  4. 430°
  5. 545°
  6. 630°
  7. 745
  8. 860

Right triangles with 30°, 45°, 60° angles

From daily life

A ladder leaning at a nice angle like 60° has a base exactly half its length, so a quick estimate needs no calculator.

Simple definition

Use the standard ratios to find sides. In a 45° triangle the legs are equal and the hypotenuse is leg × √2. In a 30°-60° triangle the sides are in the ratio 1 : √3 : 2, with the shortest side opposite 30°.

10 m5 m5√3 m60°

In the figure: A 10 m ladder at 60°: foot 10 cos 60° = 5 m, height 10 sin 60° = 5√3 m ≈ 8.66 m.

Rule 1: 30°-60° triangle — The shortest side is half the hypotenuse.

  • Easy: Hypotenuse 12: shortest side 6, longer leg 6√3.
  • Medium: Shortest side 5: hypotenuse 10, longer leg 5√3 ≈ 8.66.
  • Exam-type: An equilateral triangle of side 8 cm has height 8 sin 60° = 4√3 ≈ 6.93 cm.

Rule 2: 45° triangle — The two legs are equal.

  • Easy: Legs 7 and 7: hypotenuse 7√2 ≈ 9.9.
  • Medium: Hypotenuse 10: each leg is 10 sin 45° = 5√2 ≈ 7.07.
  • Exam-type: Check: (5√2)² + (5√2)² = 50 + 50 = 100.
Remember

Use √2 ≈ 1.41 and √3 ≈ 1.73 for decimal answers.

Common mistake

Putting the √3 side opposite 30°; the shortest side is opposite 30°.

Memory clue

"Smallest side, smallest angle: 1 faces 30°."

Practice questions

Easy
  1. 1In △ABC, right-angled at B, ∠A = 60°. Find the ratio AB/BC.
  2. 2In △ABC, right-angled at B, ∠A = 60°. Find the ratio AC/AB.
  3. 3In △ABC, right-angled at B, ∠A = 60° and BC = 7√3 cm. Find AC (in cm).
Medium
  1. 4In △ABC, right-angled at B, ∠A = 30° and BC = 8√3 cm. Find AC (in cm).
  2. 5In △ABC, right-angled at B, ∠A = 45° and BC = 5√3 cm. Find AC (in cm).
  3. 6In △ABC, right-angled at B, ∠A = 30° and AC = 2√2 cm. Find AB (in cm).
Challenging
  1. 7In △ABC, right-angled at B, ∠A = 60° and AC = 10√2 cm. Find BC (in cm).
  2. 8In △ABC, right-angled at B, ∠A = 45° and AC = 6√2 cm. Find the perimeter (in cm) of △ABC.
Show answers (try them yourself first)
  1. 11/√3
  2. 22
  3. 314
  4. 416√3
  5. 55√6
  6. 6√6
  7. 75√6
  8. 812 + 6√2