Introduction to Trigonometry — Class 10 Mathematics

Write the value of and .

Both equal .

State the identity connecting and .

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Define the trigonometric ratios sine, cosine and tangent for an acute angle.

, , and (refer to the right-triangle figure in your NCERT textbook).

Write the values of , and .

, and .

State the relationship between , and .

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State the fundamental trigonometric identity.

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Write the other two Pythagorean identities.

and .

Name the reciprocal ratios of sin, cos and tan.

, and .

If , find .

.

Evaluate .

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1. In , right-angled at , , . Determine : (i)

By Pythagoras’ theorem, . Hence, and .

1. In , right-angled at , , . Determine : (ii)

Since , and .

2. In Fig. 8.13, find .

In the right triangle, and , so . Thus, and . Therefore, .

3. If , calculate and .

Take the opposite side as and the hypotenuse as . The adjacent side is . Therefore, and .

4. Given , find and .

Here, . Take the adjacent and opposite sides as and ; then the hypotenuse is . Hence, and .

5. Given , calculate all other trigonometric ratios.

Since , take the adjacent side as and the hypotenuse as . The opposite side is . Thus, , , , , and .

6. If and are acute angles such that , then show that .

Construct right triangles having acute angles and . Since , the ratios of the adjacent side to the hypotenuse are equal. After choosing the triangles with equal hypotenuses, their corresponding adjacent sides are equal. Their remaining sides are then equal by Pythagoras’ theorem, so the triangles are congruent and hence .

7. If , evaluate : (i)

7. If , evaluate : (ii)

8. If , check whether or not.

From , and . Therefore, Taking the opposite, adjacent and hypotenuse sides as , Hence, the equality is true.

9. In triangle , right-angled at , if , find the value of: (i)

Since ,

9. In triangle , right-angled at , if , find the value of: (ii)

Since ,

10. In , right-angled at , and . Determine the values of and .

By Pythagoras’ theorem, . Thus, , so . Solving with gives and . Therefore, , , and .

11. State whether the following are true or false. Justify your answer. (i) The value of is always less than .

False. For example, .

11. State whether the following are true or false. Justify your answer. (ii) for some value of angle .

True. This means , which is possible for an acute angle because .

11. State whether the following are true or false. Justify your answer. (iii) is the abbreviation used for the cosecant of angle .

False. denotes the cosine of ; the cosecant of is written as .

11. State whether the following are true or false. Justify your answer. (iv) is the product of and .

False. is the abbreviation for the cotangent of angle , not a product.

11. State whether the following are true or false. Justify your answer. (v) for some angle .

False. For the angles considered here, , so cannot equal .

1. Evaluate the following : (i)

1. Evaluate the following : (ii)

1. Evaluate the following : (iii)

1. Evaluate the following : (iv)

1. Evaluate the following : (v)

2. Choose the correct option and justify your choice : (i) (A) (B) (C) (D)

Therefore, the correct option is (A).

2. Choose the correct option and justify your choice : (ii) (A) (B) (C) (D)

Therefore, the correct option is (D).

2. Choose the correct option and justify your choice : (iii) is true when (A) (B) (C) (D)

For , and . The other listed angles do not satisfy the equation. Therefore, the correct option is (A).

2. Choose the correct option and justify your choice : (iv) (A) (B) (C) (D)

Therefore, the correct option is (C).

3. If and ; ; , find and .

Since , gives . Also, and give . Solving, and .

4. State whether the following are true or false. Justify your answer. (i) .

False. For example, if , then , whereas .

4. State whether the following are true or false. Justify your answer. (ii) The value of increases as increases.

True for , the range considered in this chapter: increases from to .

4. State whether the following are true or false. Justify your answer. (iii) The value of increases as increases.

False. For , decreases from to as increases.

4. State whether the following are true or false. Justify your answer. (iv) for all values of .

False. They are equal at , but not for all angles; for example, while .

4. State whether the following are true or false. Justify your answer. (v) is not defined for .

True. Since and , requires division by zero and is not defined.

1. Express the trigonometric ratios and in terms of .

Let . Using , , so . Also, , and since is acute.

2. Write all the other trigonometric ratios of in terms of .

Since is acute, . From , . Therefore,

3. Choose the correct option. Justify your choice. (i) (A) (B) (C) (D)

Using , Therefore, the correct option is (B).

3. Choose the correct option. Justify your choice. (ii) (A) (B) (C) (D)

Writing everything in terms of and , The numerator is . Hence the value is . The correct option is (C).

3. Choose the correct option. Justify your choice. (iii) (A) (B) (C) (D)

Therefore, the correct option is (D).

3. Choose the correct option. Justify your choice. (iv) (A) (B) (C) (D)

Therefore, the correct option is (D).

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i)

Hence proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii)

Hence proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii) [Hint : Write the expression in terms of and ]

Hence proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iv) [Hint : Simplify LHS and RHS separately]

Also, Thus both sides are equal, and the identity is proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (v) , using the identity .

Divide the numerator and denominator on the left by : Let . Since , we have . Hence Therefore, the identity is proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vi)

Since is acute, all relevant quantities are positive. Hence proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vii)

Hence proved wherever the expressions are defined.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii)

Hence proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ix) [Hint : Simplify LHS and RHS separately]

Also, Hence proved.

4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (x)

For the first expression, For the second expression, using , Hence all three expressions are equal.