Chapter 8 of CBSE NCERT Class 10 Math covers Trigonometry. Concepts covered in Chapter 8 include trigonometric ratios, trigonometric ratios of complementary angles, trigonometric identities. The extra questions given below include questions akin to HOTS (Higher Order Thinking Skills) questions and exemplar questions of NCERT.
Here is a quick recap of the key concepts that are covered in this chapter in the CBSE NCERT Class 10 Math text book.
The trigonometric ratios of the angle A in right triangle ABC are defined as follows:
sine of ∠A = \\frac{\text{side oppposite to angle A}}{\text{hypotenuse}}\\) = \\frac{\text{BC}}{\text{AC}}\\)
cosine of ∠A = \\frac{\text{side adjacent to angle A}}{\text{hypotenuse}}\\) = \\frac{\text{AB}}{\text{AC}}\\)
tangent of ∠A = \\frac{\text{side oppposite to angle A}}{\text{side adjacent to angle A}}\\) = \\frac{\text{BC}}{\text{AB}}\\)
The ratios of cosec A, sec A and cot A are the reciprocals of sin A, cos A and tan A respectively.
cosecant of ∠A = \\frac{\text{1}}{\text{sine of angle A}}\\) = \\frac{\text{AC}}{\text{BC}}\\)
secant of ∠A = \\frac{\text{1}}{\text{cosine of angle A}}\\) = \\frac{\text{AC}}{\text{AB}}\\)
cotangent of ∠A = \\frac{\text{1}}{\text{tangent of angle A}}\\) = \\frac{\text{AB}}{\text{BC}}\\)
The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the traiangle, if the angle remains the same.
∠A | 0° | 30° | 45° | 60° | 90° |
---|---|---|---|---|---|
sin A | 0 | \\frac{1}{2}\\) | \\frac{1}{{\sqrt{2}}}\\) | \\frac{{\sqrt{3}}}{2}\\) | 1 |
cos A | 1 | \\frac{{\sqrt{3}}}{2}\\) | \\frac{1}{{\sqrt{2}}}\\) | \\frac{1}{2}\\) | 0 |
tan A | 0 | \\frac{1}{{\sqrt{3}}}\\) | 1 | √3 | Not defined |
cosec A | Not defined | 2 | √2 | \\frac{2}{{\sqrt{3}}}\\) | 1 |
sec A | 1 | \\frac{2}{{\sqrt{3}}}\\) | √2 | 2 | Not defined |
cot A | Not defined | √3 | 1 | \\frac{1}{{\sqrt{3}}}\\) | 0 |
∴ The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than or equal to 1
The two angles are said to be complementary if their sum equals 90°.
The trigonometric ratios of complementary angles are as follows:
An equation involving trigonometric ratios of an angle is said to be trigonometric idnetity, if it is true for all values of the angles involved.
Some of the key trigonometric identities used in this chapter are as follows:
Question 1
Consider right triangle ABC, right angled at B. If AC = 17 units and BC = 8 units determine all the trigonometric ratios of angle C.
Question 2
Question 3
Question 4
In ΔABC right angled at B, sin C = \\frac{\text{5}}{\text{13}}). Find :
(i) sin A
(ii) cos A
(iii) cos C
Question 5
In triangle ABC, right angled at B if sin A = \\frac{\text{1}}{\text{2}}), find the value of
1. sin C cos A – cos C sin A
2. cos A cos C + sin A sin C
Question 6
Question 7
In triangle ABC right angled at B, AB = 12cm and ∠CAB = 60°. Determine the lengths of the other two sides.
Question 8
Question 9
Question 10
Find the value of θ in each of the following. θ is an acute angle.
(i) 3 sec 2θ = 2√3
(ii) 4 cot 3θ - 4 = 0
(iii) 2 sin 2θ = 1
Question 11
Solve the following: 0 < θ < 90°
(i) 2 sin^{2} θ = \\frac{3}{2})
(ii) 3 tan^{2} θ + 2 = 3
(iii) cos^{2} θ - \\frac{1}{4})= \\frac{1}{2})
Question 12
Evaluate the following:
(i) cosec^{2} 45° + tan^{2} 45° – 3sin^{2} 90°
(ii) cos 60° cos 30° – sin 60° sin 30°
(iii) \\frac{\text{tan 60° − tan 30°}}{\text{1 +tan 60° tan 30°}})
Question 13
Find the value of x in each of the following:
(i) cosec 3x = \\frac{\text{cot 30° + cot 60°}}{\text{1 + cot 30° cot 60°}})
(ii) cos x = 2 sin 45° cos 45° – sin 30°
Question 14
If θ is an acute angle and \\frac{\text{sinθ + 1 }}{\text{sinθ − 1}}) =\\frac{\text{√3 + 2 }}{\text{√3 − 2 }}), find θ.
Question 15
In an acute angle triangle ABC, sin (A + B - C) = \\frac{\text{1}}{\text{2}}), cot (A - B + C) = 0 and cos (B + C – A) =\\frac{\text{1}}{\text{2}}). What are the values of A, B, and C?
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