In a right triangle $\triangle ABC$, with $\angle B = 90^\circ$ and $\angle A = 60^\circ$, if the hypotenuse $AC$ is 10 cm, what is the length of the side $BC$?

Correct Option: B. Solution: In a right triangle, the side opposite $60^\circ$ is $\frac{\sqrt{3}}{2}$ of the hypotenuse. Thus, $BC = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}$ cm.

Introduction to Trigonometry

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Summary

Summary of Introduction to Trigonometry

Key Concepts

Trigonometric Ratios: Defined for acute angles in a right triangle.
- Sine (sin A): Opposite side / Hypotenuse
- Cosine (cos A): Adjacent side / Hypotenuse
- Tangent (tan A): Opposite side / Adjacent side
- Cosecant (cosec A): 1 / sin A
- Secant (sec A): 1 / cos A
- Cotangent (cot A): 1 / tan A

Important Values for Specific Angles

0°: sin 0° = 0, cos 0° = 1, tan 0° = 0
30°: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
45°: sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
90°: sin 90° = 1, cos 90° = 0, tan 90° = Not defined

Trigonometric Identities

sin² A + cos² A = 1
1 + tan² A = sec² A
1 + cot² A = cosec² A

Applications

Used to find heights and distances in real-world scenarios (e.g., measuring heights of buildings, distances across rivers).

Common Misconceptions

The value of sin A or cos A never exceeds 1.
cot A is not defined for A = 0°.

Exercises

Evaluate trigonometric expressions and verify identities as practice.

Learning Objectives

Understand the definition and significance of trigonometric ratios in right triangles.
Identify and calculate the sine, cosine, tangent, cosecant, secant, and cotangent ratios for given angles.
Apply trigonometric identities to simplify expressions and solve equations.
Evaluate trigonometric ratios for specific angles (0°, 30°, 45°, 60°, 90°).
Prove trigonometric identities and understand their applications in solving problems.
Analyze the relationships between the sides and angles of right triangles using trigonometric concepts.
Justify the truth or falsehood of statements related to trigonometric functions and their properties.

Detailed Notes

Introduction to Trigonometry

Trigonometric Ratios

In a right triangle ABC, right-angled at B:

Sine:
- sin A = side opposite to angle A / hypotenuse = AC / BC
Cosine:
- cos A = side adjacent to angle A / hypotenuse = AB / AC
Tangent:
- tan A = side opposite to angle A / side adjacent to angle A = AB / BC
Cosecant:
- cosec A = 1/sin A = hypotenuse / side opposite to angle A = AC / AB
Secant:
- sec A = 1/cos A = hypotenuse / side adjacent to angle A = AC / AB
Cotangent:
- cot A = 1/tan A = side adjacent to angle A / side opposite to angle A = BC / AB

Trigonometric Ratios of Specific Angles

Angle (A)	0°	30°	45°	60°	90°
sin A	0	1/2	√2/2	√3/2	1
cos A	1	√3/2	√2/2	1/2	0
tan A	0	1/√3	1	√3	Not defined
cosec A	Not defined	2	√2	2/√3	1
sec A	1	2/√3	√2	2	Not defined
cot A	Not defined	√3	1	1/√3	0

Trigonometric Identities

Pythagorean Identity:
- cos² A + sin² A = 1
Tangent and Secant Identity:
- 1 + tan² A = sec² A
Cosecant and Cotangent Identity:
- 1 + cot² A = cosec² A

Exercises

Exercise 8.1

In triangle ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine:
- (i) sin A, cos A
- (ii) sin C, cos C
If sin A = 4, calculate cos A and tan A.
Given 15 cot A = 8, find sin A and sec A.

Exercise 8.2

Evaluate the following:
- (i) sin 60° cos 30° + sin 30° cos 60°
- (ii) 2 tan² 45° + cos² 30° - sin² 60°
- (iii) sec 30° + cosec 30°
- (iv) sin 30° + tan 45° - cosec 60°
- (v) 5 cos² 60° + 4 sec² 30° - tan² 45°
Choose the correct option and justify your choice for various equations.

True or False Statements

sin (A+B) = sin A + sin B.
The value of sin Θ increases as Θ increases.
cot A is not defined for A = 0°.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Trigonometry

Common Pitfalls

Misunderstanding Trigonometric Ratios: Students often confuse the definitions of sine, cosine, and tangent. Remember:
- sin A = opposite/hypotenuse
- cos A = adjacent/hypotenuse
- tan A = opposite/adjacent
Incorrectly Applying Identities: Students may misapply identities like sin²A + cos²A = 1. Ensure you understand the conditions under which these identities hold.
Ignoring Angle Restrictions: Some trigonometric functions are not defined for certain angles (e.g., cot A is not defined for A = 0°). Always check the domain of the functions you are using.
Confusing Degrees and Radians: Ensure you are consistent with the units you are using. Misinterpretation can lead to incorrect answers.
Neglecting to Justify Answers: When asked to justify whether a statement is true or false, provide a clear explanation based on definitions or identities.

Exam Tips

Practice with Specific Angles: Familiarize yourself with the values of trigonometric ratios for angles 0°, 30°, 45°, 60°, and 90°. This will help you solve problems more quickly.
Use Diagrams: Whenever possible, draw diagrams to visualize the problem. This can help you understand the relationships between the angles and sides.
Check Your Work: If time permits, go back and check your answers, especially for calculations involving multiple steps.
Understand the Context: In word problems, identify the right triangle and the relevant sides and angles before applying trigonometric ratios.
Memorize Key Identities: Knowing key identities like sec²A = 1 + tan²A can save time during exams.

Practice & Assessment

Multiple Choice Questions

4 cm

4\sqrt{2} cm

8 cm

8\sqrt{3} cm

Correct Answer: C

Solution:

In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Thus,

\cos 60^\circ = \frac{4}{AC}

. Since

\cos 60^\circ = \frac{1}{2}

, we have

\frac{1}{2} = \frac{4}{AC}

. Solving for

AC

, we get

AC = 8

cm.

5 cm

5√3 cm

10 cm

10√3 cm

Correct Answer: B

Solution:

In a right triangle, the side opposite a 60° angle is

\frac{\sqrt{3}}{2}

times the hypotenuse. Therefore, the length of the leg is

\frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3}

cm.

3√3 cm

6√3 cm

3 cm

6 cm

Correct Answer: C

Solution:

In a 30° angle triangle, the ratio of the opposite to the adjacent side is

\tan 30° = \frac{1}{\sqrt{3}}

. Therefore, the opposite side is

6 \times \frac{1}{\sqrt{3}} = 3

cm.

10 cm

5√3 cm

7 cm

6 cm

Correct Answer: A

Solution:

In a 30°-60°-90° triangle, the hypotenuse is twice the length of the side opposite the 30° angle. Therefore,

AC = 2 \times 5 = 10

cm.

60°

30°

45°

90°

Correct Answer: A

Solution:

In a triangle, the sum of angles is 180°. Given angles B (90°) and C (30°), angle A must be 60°.

a\sqrt{2}

2a

a

a\sqrt{3}

Correct Answer: A

Solution:

In a

45^\circ

45^\circ

90^\circ

triangle, the hypotenuse is

\sqrt{2}

times the length of each leg. Thus, the hypotenuse is

a\sqrt{2}

5 cm

10 cm

5√3 cm

10√3 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the hypotenuse is twice the length of the side opposite the 30° angle. Therefore, the hypotenuse is 10 cm.

8√3 cm

16 cm

8√2 cm

16√3 cm

Correct Answer: A

Solution:

In a 30°-60°-90° triangle, the side adjacent to the 30° angle is half the hypotenuse. Therefore, the hypotenuse is twice the length of the side adjacent to the 30° angle. Hence, the hypotenuse is 8√3 cm.

5 cm

5\sqrt{3} cm

10 cm

10\sqrt{3} cm

Correct Answer: A

Solution:

In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse. Thus,

\sin 30^\circ = \frac{BC}{10}

. Since

\sin 30^\circ = \frac{1}{2}

, we have

\frac{1}{2} = \frac{BC}{10}

. Solving for

BC

, we get

BC = 5

cm.

12 cm

10 cm

11 cm

13 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem,

BC = \sqrt{AC^2 - AB^2} = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12

cm.

12√3 cm

6 cm

4√3 cm

4 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the side adjacent to the 60° angle. Therefore, the side adjacent to the 60° angle is 12/√3 = 4√3 cm.

12 cm

10 cm

8 cm

9 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem, let the other leg be

x

. Then,

13^2 = 5^2 + x^2 \Rightarrow 169 = 25 + x^2 \Rightarrow x^2 = 144 \Rightarrow x = 12

cm.

7√2 cm

14 cm

7 cm

14√2 cm

Correct Answer: A

Solution:

In a 45°-45°-90° triangle, the hypotenuse is √2 times the length of either leg. Thus, if the leg is 7 cm, the hypotenuse is 7√2 cm.

7 cm

7√2 cm

14 cm

14√2 cm

Correct Answer: B

Solution:

In a 45°-45°-90° triangle, the hypotenuse is √2 times the length of either leg. Therefore, the hypotenuse is 7√2 cm.

15 cm

7.5 cm

15√3 cm

7.5√3 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side opposite the 30° angle is half the hypotenuse. Therefore, the opposite side is 15/2 = 7.5 cm.

10 cm

12 cm

14 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem,

c = \sqrt{a^2 + b^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

cm.

9√3 cm

18 cm

9 cm

9√2 cm

Correct Answer: A

Solution:

In a 30°-60°-90° triangle, the hypotenuse is twice the length of the side opposite the 30° angle. Since the side opposite the 60° angle is 9 cm, the hypotenuse is 9√3 cm.

7\sqrt{2} cm

14 cm

7 cm

14\sqrt{2} cm

Correct Answer: A

Solution:

In a

45^\circ-45^\circ-90^\circ

triangle, the hypotenuse is

\sqrt{2}

times the length of each leg. Therefore,

AC = 7\sqrt{2}

cm.

8√3 cm

4√3 cm

8 cm

4 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the side adjacent to it. Therefore, the opposite side is 8√3/2 = 4√3 cm.

5 cm

10 cm

8.66 cm

7.5 cm

Correct Answer: B

Solution:

In a right triangle, the side opposite a

30^\circ

angle is half the hypotenuse. Therefore, if the side opposite the

30^\circ

angle is 5 cm, the hypotenuse

AC

2 \times 5 = 10

cm.

5 cm

5\sqrt{3} cm

10\sqrt{3} cm

10 cm

Correct Answer: B

Solution:

In a right triangle, the side opposite

60^\circ

\frac{\sqrt{3}}{2}

of the hypotenuse. Thus,

BC = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}

cm.

a\sqrt{2}

a\sqrt{3}

2a

a

Correct Answer: A

Solution:

In a right triangle with a

45^\circ

angle, the sides opposite and adjacent to the angle are equal, and the hypotenuse is

\sqrt{2}

times the length of either leg. Therefore, the hypotenuse is

a\sqrt{2}

8 cm

4 cm

7 cm

5 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem:

c^2 = a^2 + b^2

, where

c = 10

and

a = 6

, solve for

b

b = \sqrt{10^2 - 6^2} = 8

cm.

10 cm

5√3 cm

15 cm

5√2 cm

Correct Answer: A

Solution:

In a right triangle with a 30° angle, the side opposite to the 30° angle is half the hypotenuse. Thus, if the opposite side is 5 cm, the hypotenuse is 10 cm.

7√2 cm

14 cm

7 cm

14√2 cm

Correct Answer: A

Solution:

In a 45°-45°-90° triangle, the hypotenuse is

\sqrt{2}

times the length of each leg. Thus, the hypotenuse is

7\sqrt{2}

cm.

10 cm

12 cm

14 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem,

AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

cm.

13 cm

10 cm

14 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem, hypotenuse

AC = \sqrt{(AB)^2 + (BC)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

cm.

4 cm

4√3 cm

4√2 cm

2√3 cm

Correct Answer: C

Solution:

Using the Pythagorean theorem, the other leg is calculated as √(8² - 4²) = √(64 - 16) = √48 = 4√3 cm.

10 cm

5√3 cm

5 cm

10√3 cm

Correct Answer: A

Solution:

In a 30°-60°-90° triangle, the hypotenuse is twice the length of the side opposite the 30° angle.

7 cm

9.9 cm

7\sqrt{2}$ cm

14 cm

Correct Answer: C

Solution:

In a

45^\circ-45^\circ-90^\circ

triangle, the hypotenuse is

\sqrt{2}

times the length of each leg. Therefore,

AC = 7\sqrt{2}

cm.

5√2 cm

5 cm

10 cm

7 cm

Correct Answer: A

Solution:

In a right triangle with a 45° angle, the sides opposite and adjacent to the angle are equal. Using

\sin 45° = \frac{1}{\sqrt{2}}

, the length of the opposite side is

10 \times \frac{1}{\sqrt{2}} = 5\sqrt{2}

cm.

5 cm

5√3 cm

10 cm

10√3 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the side opposite the 30° angle. Given the hypotenuse is 10 cm, the side opposite the 60° angle is 5√3 cm.

\frac{3}{5}

\frac{4}{5}

\frac{3}{4}

\frac{4}{3}

Correct Answer: A

Solution:

In a right triangle, sine of an angle is the ratio of the opposite side to the hypotenuse. Here,

AB

is opposite

\angle BAC

and

AC

is the hypotenuse. Using the Pythagorean theorem,

AC = \sqrt{6^2 + 8^2} = 10

cm. Thus,

\sin \angle BAC = \frac{AB}{AC} = \frac{6}{10} = \frac{3}{5}

9 cm

9√2 cm

18 cm

18√2 cm

Correct Answer: A

Solution:

In a 45°-45°-90° triangle, the sides opposite the 45° angles are equal. Therefore, if the side adjacent to a 45° angle is 9 cm, the other side is also 9 cm.

5 cm

9 cm

10 cm

11 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem:

c^2 = a^2 + b^2

, where

c = 13

cm and

a = 12

cm. Solving for

b

b^2 = 13^2 - 12^2 = 169 - 144 = 25

, thus

b = 5

cm.

12 cm

6√3 cm

6 cm

12√3 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side adjacent to the 30° angle is half the hypotenuse. Therefore, the hypotenuse is 6√3 cm.

9√2 cm

18 cm

9 cm

18√2 cm

Correct Answer: A

Solution:

In a 45°-45°-90° triangle, the hypotenuse is √2 times the length of either leg.

6 cm

6√2 cm

12 cm

12√2 cm

Correct Answer: B

Solution:

In a 45°-45°-90° triangle, the hypotenuse is √2 times the length of either leg. Therefore, the hypotenuse is 6√2 cm.

\sqrt{3}:1

1:\sqrt{3}

1:2

2:1

Correct Answer: A

Solution:

In a

30^\circ-60^\circ-90^\circ

triangle, the sides are in the ratio

1: \sqrt{3}: 2

. Therefore, the side opposite

60^\circ

(which is

BC

) is

\sqrt{3}

times the side opposite

30^\circ

(which is

AB

10√3 cm

20 cm

10 cm

20√3 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side adjacent to the 30° angle is √3/2 times the hypotenuse. Therefore, the hypotenuse is 10 * 2/√3 = 20 cm.

8√3 cm

16 cm

8 cm

16√3 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the side opposite the 30° angle. The hypotenuse is twice the side opposite the 30° angle, so it is 16 cm.

2√3 cm

4√3 cm

8 cm

4 cm

Correct Answer: B

Solution:

In a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the length of the side adjacent to the 60° angle. Therefore, it is 4√3 cm.

4\sqrt{3} cm

8 cm

8\sqrt{3} cm

4 cm

Correct Answer: B

Solution:

For a

60^\circ

angle in a right triangle, the side adjacent is half the hypotenuse. Therefore, the hypotenuse is

2 \times 4 = 8

cm.

8 cm

10 cm

12 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem:

c^2 = a^2 + b^2

. Here,

13^2 = 5^2 + b^2

. Solving gives

b = 8

cm.

5 cm

10 cm

5√3 cm

10√3 cm

Correct Answer: B

Solution:

In a right-angled triangle with a 30° angle, the side opposite the angle is half the hypotenuse. Therefore, if BC = 5 cm, then AC = 2 * BC = 10 cm.

10 cm

5\sqrt{3} cm

5\sqrt{2} cm

15 cm

Correct Answer: A

Solution:

In a right triangle with an angle of

30^\circ

, the side opposite this angle is half the hypotenuse. Therefore, if the side opposite is 5 cm, the hypotenuse is

2 \times 5 = 10

Solution:

The diagram description for Fig. 8.4 states that the hypotenuse is the side opposite the right angle, connecting vertices A and C.

Introduction to Trigonometry

AI Learning Assistant

Summary

Summary of Introduction to Trigonometry

Key Concepts

Important Values for Specific Angles

Trigonometric Identities

Applications

Common Misconceptions

Exercises

Learning Objectives

Learning Objectives

Detailed Notes

Introduction to Trigonometry

Trigonometric Ratios

Trigonometric Ratios of Specific Angles

Trigonometric Identities

Exercises

Exercise 8.1

Exercise 8.2

True or False Statements

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Trigonometry

Common Pitfalls

Exam Tips

Practice & Assessment

Multiple Choice Questions

Q1.In a right triangle, if the angle at vertex AAA is 60∘60^\circ60∘ and the side adjacent to this angle is 4 cm, what is the length of the hypotenuse ACACAC?

Correct Answer: C

Q2.In a right triangle, if the hypotenuse is 10 cm and the angle opposite to one leg is 60°, what is the length of that leg?

Correct Answer: B

Q3.In a right triangle, if the side adjacent to a 30° angle is 6 cm, what is the length of the opposite side?

Correct Answer: C

Q4.In a right triangle, if angle ACB is 30° and the side opposite this angle (AB) is 5 cm, what is the length of the hypotenuse (AC)?

Correct Answer: A

Q5.In a right triangle, if the angle at vertex B is 90° and the angle at vertex C is 30°, what is the measure of angle A?

Correct Answer: A

Q6.Given a right triangle with vertices A, B, and C, where the angle at B is 90∘90^\circ90∘ and angle ∠BAC=45∘\angle BAC = 45^\circ∠BAC=45∘, if the side opposite to ∠BAC\angle BAC∠BAC is aaa, what is the length of the hypotenuse?

Correct Answer: A

Q7.In a right-angled triangle, if the angle at vertex C is 30°, and the side opposite to this angle is 5 cm, what is the length of the hypotenuse?

Correct Answer: B

Q8.In a right triangle with a 30° angle at vertex C, if the side adjacent to this angle is 8 cm, what is the length of the hypotenuse?

Correct Answer: A

Q9.In a right triangle, if the hypotenuse AC=10AC = 10AC=10 cm and angle ∠ACB=30∘\angle ACB = 30^\circ∠ACB=30∘, what is the length of the side opposite to ∠ACB\angle ACB∠ACB?

Correct Answer: A

Q10.In a right triangle with a right angle at BBB, if AB=9AB = 9AB=9 cm and the hypotenuse AC=15AC = 15AC=15 cm, what is the length of BCBCBC?

Correct Answer: A

Q11.In a right triangle, if the side opposite to a 60° angle is 12 cm, what is the length of the side adjacent to the 60° angle?

Correct Answer: B

Q12.In a right triangle, if the hypotenuse is 13 cm and one leg is 5 cm, what is the length of the other leg?

Correct Answer: A

Q13.Consider a right triangle where the side opposite to angle 45° is 7 cm. What is the length of the hypotenuse?

Correct Answer: A

Q14.In a right triangle, the side adjacent to a 45° angle is 7 cm. What is the length of the hypotenuse?

Correct Answer: B

Q15.If a right triangle has a hypotenuse of 15 cm and one of the angles is 30°, what is the length of the side opposite to the 30° angle?

Correct Answer: B

Q16.In a right triangle with sides aaa, bbb, and hypotenuse ccc, if a=6a = 6a=6 cm and b=8b = 8b=8 cm, what is the length of the hypotenuse ccc?

Correct Answer: A

Q17.In a right triangle, if the side opposite to the 60° angle is 9 cm, what is the length of the hypotenuse?

Correct Answer: A

Q18.In a right triangle △ABC\triangle ABC△ABC, with ∠B=90∘\angle B = 90^\circ∠B=90∘ and ∠A=45∘\angle A = 45^\circ∠A=45∘, if the side AB=7AB = 7AB=7 cm, what is the length of the hypotenuse ACACAC?

Correct Answer: A

Q19.In a right triangle, if the side adjacent to a 60° angle is 8 cm, what is the length of the opposite side?

Correct Answer: B

Q20.In a right triangle, if angle ∠BAC=30∘\angle BAC = 30^\circ∠BAC=30∘ and the side opposite this angle is 5 cm, what is the length of the hypotenuse ACACAC?

Correct Answer: B

Q21.In a right triangle △ABC\triangle ABC△ABC, with ∠B=90∘\angle B = 90^\circ∠B=90∘ and ∠A=60∘\angle A = 60^\circ∠A=60∘, if the hypotenuse ACACAC is 10 cm, what is the length of the side BCBCBC?

Correct Answer: B

Q22.In a right triangle, if the angle at vertex AAA is 45∘45^\circ45∘ and the side opposite to it is aaa, what is the length of the hypotenuse?

Correct Answer: A

Q23.In a right triangle, if the hypotenuse is 10 cm and one leg is 6 cm, what is the length of the other leg?

Correct Answer: A

Q24.In a right triangle, if the side opposite to a 30° angle is 5 cm, what is the length of the hypotenuse?

Correct Answer: A

Q25.In a right triangle, if angle BAC is 45° and the side adjacent to this angle is 7 cm, what is the length of the hypotenuse?

Correct Answer: A

Q26.In a right triangle with a right angle at vertex B, if AB = 6 cm and BC = 8 cm, what is the length of the hypotenuse AC?

Correct Answer: A

Q27.In a right triangle, if the angle at vertex B is 90° and the side opposite to angle C is 5 cm, what is the length of the hypotenuse if the side adjacent to angle C is 12 cm?