Introduction to Trigonometry Class 10 Previous Year Questions (Chapter 8)

1. 1

   Step 1: Write the standard values

   ½ mark

   sin 45° = 1/√2, sec 30° = 2/√3, tan 30° = 1/√3.

   Why? The trigonometric ratios of 30°, 45° and 60° have fixed standard values, so they can be substituted directly into the expression.

2. 2

   Step 2: Simplify the expression

   1½ marks

   \(\frac{\sin 45^\circ}{\sec 30^\circ - \tan 30^\circ} = \frac{1/\sqrt{2}}{2/\sqrt{3} - 1/\sqrt{3}} = \frac{1/\sqrt{2}}{1/\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{6}}{2}.\) Therefore, the value = √6/2.

   Why? The two terms in the denominator are like surds, so they combine into one term. Dividing by a fraction multiplies by its reciprocal, and rationalising the denominator gives the simplest form.

Marks: ½ + 1½ = 2 · Split as in the CBSE marking scheme, 2025 · Basic 430/2/2 Q22 (the marks split is taken from the 2025 Basic paper, where this question also appeared)

1. 1

   Step 1: Write the standard values

   1 mark (suggested)

   sin 30° = 1/2, cos 60° = 1/2, cos 30° = √3/2, sin 60° = √3/2, cot 45° = 1.

   Why? These are the trigonometric ratios of the specific angles given in Table 8.1. Substituting them changes the trigonometric expression into numbers.

2. 2

   Step 2: Evaluate the expression

   1 mark (suggested)

   (1/2)(1/2) + (√3/2)(√3/2) − 1 = 1/4 + 3/4 − 1 = 0.

   Why? Multiply the fractions first, then add the two products and subtract. The two products together make 1, and the subtracted value is also 1, so they cancel and the result is 0.

Marks: 1 + 1 = 2 · Suggested split (CBSE’s usual pattern for 2-mark questions)

1. 1

   Step 1: Substitute the standard values

   1 mark (suggested)

   5(√3/2)² + 3(√3/2)² − (√2)²

   Why? The angles 60°, 30° and 45° are specific angles whose trigonometric ratios have fixed exact values; substituting those values turns the expression into an arithmetic one.

2. 2

   Step 2: Simplify

   1 mark (suggested)

   5(3/4) + 3(3/4) − 2 = 15/4 + 9/4 − 2 = 4. The value of the expression is 4.

   Why? Squaring each ratio first and then adding and subtracting in order gives a single number, which is the value of the expression.

Marks: 1 + 1 = 2 · Suggested split (CBSE’s usual pattern for 2-mark questions)

1. 1

   Step 1: Substitute the standard values

   1 mark (suggested)

   x = sin 60° cos 30° + cos 60° sin 30° = (√3/2)(√3/2) + (1/2)(1/2).

   Why? The trigonometric ratios of 30° and 60° have fixed standard values; replacing sin 60°, cos 30°, cos 60° and sin 30° by those values changes the expression into ordinary fractions.

2. 2

   Step 2: Simplify to get x

   1 mark (suggested)

   x = 3/4 + 1/4 = 1. Hence, x = 1.

   Why? Multiplying the fractions gives 3/4 and 1/4, and adding them gives the final value of x.

Marks: 1 + 1 = 2 · Suggested split (CBSE’s usual pattern for 2-mark questions)

1. 1

   Step 1: Substitute the standard values and simplify

   1½ marks

   sin²60° + cos²60° = (√3/2)² + (1/2)² = 3/4 + 1/4 = 1; tan60° = √3, tan30° = 1/√3; so (5√3)/(1 × 1/√3) = 5√3 × √3 = 15.

   Why? The identity sin²A + cos²A = 1 collapses the bracket to 1, and the standard values of tan60° and tan30° turn the denominator into a single exact number.

2. 2

   Step 2: State the final value

   ½ mark

   Hence, the value of the expression is 15.

   Why? A fraction divided by a unit fraction is simplified by multiplying by its reciprocal, which gives the exact whole number.

Marks: 1½ + ½ = 2 · Split as in the CBSE marking scheme, 2024 · Standard 30/3/3 Q22