NCERT Solutions Class 10 Maths Chapter 8 — Introduction to Trigonometry

Solution:

Step 1: Identify the sides relative to angle A. The side opposite to angle A is BC = 6 cm. The side adjacent to angle A is AB = 8 cm.

Step 2: Apply the definition of tan A. [tan A = Opposite / Adjacent]

Step 3: Substitute the given values. [tan A = BC / AB = 6 / 8]

Step 4: Simplify the fraction. [tan A = 3 / 4]

Solution:

Step 1: For an acute angle A, sin A and cos A are ratios of a leg to the hypotenuse. Since the hypotenuse is always the longest side, sin A and cos A are always less than or equal to 1.

Step 2: sec A is the reciprocal of cos A. Since cos A ≤ 1, sec A (1/cos A) must be greater than or equal to 1 for an acute angle.

Step 3: tan A is the ratio of the opposite side to the adjacent side. In a right-angled triangle, the opposite side can be longer than the adjacent side, or vice-versa. Therefore, tan A can be less than 1, equal to 1 (for 45°), or greater than 1.

Step 4: cos A = Adjacent / Hypotenuse. Since Adjacent < Hypotenuse, cos A must always be less than 1. So, 5/3 (which is > 1) is not possible for cos A.

Solution:

Step 1: The cosecant of an angle is the reciprocal of its sine.

Step 2: Write down the reciprocal relationship. [cosec θ = 1 / sin θ]

Step 3: Substitute the given value of sin θ. [cosec θ = 1 / (12/13)]

Step 4: Simplify the expression. [cosec θ = 13 / 12]

Solution:

Step 1: Recall the standard values: sin 30° = 1/2 and tan 45° = 1.

Step 2: Substitute these values into the expression. [2 × sin 30° + tan 45° = 2 × (1/2) + 1]

Step 3: Perform the multiplication. [ = 1 + 1]

Step 4: Perform the addition. [ = 2]

Solution:

Step 1: In a right-angled triangle PQR, right-angled at Q, PR is the hypotenuse.

Step 2: For angle P, the side adjacent to it is PQ.

Step 3: The definition of cos P is Adjacent / Hypotenuse.

Step 4: Therefore, cos P = PQ / PR. Ravi's formula is correct.