Unit 6: Introduction to Trigonometry

Exercise 6.2 Solutions

Step-by-step solved textbook exercises for right-angled triangles, finding six basic trigonometric ratios, complementary angles, and verification of reciprocal identities.

Question 1 Ratio Calculation
For each of the following right-angled triangles, find the six trigonometric ratios of θ (and the remaining acute angle φ):
  • (a) Perpendicular = 4, Base = 3, Hypotenuse = 5.
  • (b) Perpendicular = 8, Base = 15, Hypotenuse = 17.
  • (c) Perpendicular = 5, Base = 12, Hypotenuse = 13.
Solution

The six trigonometric ratios for an angle θ are defined as:

sin θ = PerpHyp
cos θ = BaseHyp
tan θ = PerpBase
cosec θ = HypPerp
sec θ = HypBase
cot θ = BasePerp

Case (a): Perpendicular = 4, Base = 3, Hypotenuse = 5:

Angle sin cos tan cosec sec cot
θ 45 35 43 54 53 34
φ (other angle) 35 45 34 53 54 43

Case (b): Perpendicular = 8, Base = 15, Hypotenuse = 17:

Angle sin cos tan cosec sec cot
θ 817 1517 815 178 1715 158
φ (other angle) 1517 817 158 1715 178 815

Case (c): Perpendicular = 5, Base = 12, Hypotenuse = 13:

Angle sin cos tan cosec sec cot
θ 513 1213 512 135 1312 125
φ (other angle) 1213 513 125 1312 135 512
Question 2 Variables Ratios
For a right-angled triangle ABC, find the trigonometric ratios for which m∠A = θ and m∠C = 90°. Let the side lengths be BC = a, AC = b, and AB = c.
Solution

Since the right angle is at C (m∠C = 90°):

  • Hypotenuse is the side AB = c.
  • For angle A = θ: Perpendicular is BC = a and Base is AC = b.

The trigonometric ratios for θ are:

  • sin θ = OppositeHypotenuse = ac
  • cos θ = AdjacentHypotenuse = bc
  • tan θ = OppositeAdjacent = ab
Question 3 Correction Applied
Considering the adjoining triangle ABC (with side lengths: perpendicular = 3, base = 4, hypotenuse = 5), verify that:
  • (i) sin θ × cosec θ = 1
  • (ii) cos θ × sec θ = 1
  • (iii) tan θ × cot θ = 1
Solution

From the right-angled triangle ABC:

  • sin θ = 35,   cosec θ = 53
  • cos θ = 45,   sec θ = 54
  • tan θ = 34,   cot θ = 43
  • (i) Verify sin θ × cosec θ = 1:
    L.H.S = sin θ × cosec θ = 35 × 53 = 1 = R.H.S. (Verified)
  • (ii) Verify cos θ × sec θ = 1:
    L.H.S = cos θ × sec θ = 45 × 54 = 1 = R.H.S. (Verified)
  • (iii) Verify tan θ × cot θ = 1:
    L.H.S = tan θ × cot θ = 34 × 43 = 1 = R.H.S. (Verified)
Note on Textbook Typo The textbook intermediate solution details write `sin8.secO = 1` by mistake (combining sin and sec). We have corrected this to state the proper reciprocal identity sin θ × cosec θ = 1.
Question 4 Complementary Identities
Fill in the blanks:
(i) sin 30° = cos _____
(ii) cos 30° = sin _____
(iii) tan 30° = cot _____
(iv) tan 60° = cot _____
(v) sin 60° = cos _____
(vi) cos 60° = sin _____
(vii) sin 45° = cos _____
(viii) tan 45° = cot _____
(ix) cos 45° = sin _____
Solution

These fill-in-the-blank items demonstrate the co-function (complementary angle) identities: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, and tan(90° − θ) = cot θ.

  • (i) sin 30° = cos (90° − 30°) = cos 60°
  • (ii) cos 30° = sin (90° − 30°) = sin 60°
  • (iii) tan 30° = cot (90° − 30°) = cot 60°
  • (iv) tan 60° = cot (90° − 60°) = cot 30°
  • (v) sin 60° = cos (90° − 60°) = cos 30°
  • (vi) cos 60° = sin (90° − 60°) = sin 30°
  • (vii) sin 45° = cos (90° − 45°) = cos 45°
  • (viii) tan 45° = cot (90° − 45°) = cot 45°
  • (ix) cos 45° = sin (90° − 45°) = sin 45°
Question 5 Correction Applied
If in a right-angled triangle ABC, m∠B = 90° and C is an acute angle of 60°. Let the side lengths opposite to A, B, and C be a, b (hypotenuse), and c respectively. Find all ten trigonometric ratios for the acute angles (60° and 30°).
Solution

Since m∠B = 90° and m∠C = 60°, the third angle is m∠A = 90° − 60° = 30°.

The hypotenuse is the side opposite to B, which is b.

1. Ratios for Angle C = 60°:

Perpendicular = c, Base = a, Hypotenuse = b.

sin 60° = cb
cos 60° = ab
tan 60° = ca
cosec 60° = bc
cot 60° = ac

2. Ratios for Angle A = 30°:

Perpendicular = a, Base = c, Hypotenuse = b.

sin 30° = ab
cos 30° = cb
tan 30° = ac
sec 30° = bc
cot 30° = ca
Note on Textbook Wording The textbook question contains a typo in its statement `sin mZA = ...` which is incomplete. However, the solved key provides the full ten trigonometric ratios shown above for both the 30° and 60° acute angles. We solved the complete set here.

🎲 Exercise 6.2 Trigonometric Ratio Playground

Adjust the side lengths of a right triangle to dynamically compute all trigonometric ratios and verify Reciprocal & Quotient identities!

Angle θ (at bottom)

Angle φ (at top)