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Example 1. Express 1·2 radians in degree measure.
1·2 radians =
Example 2. Calculate sin α if cos α = - 9/11 and α ∈ (π, 3π/2).
For any angle a belonging to the indicated interval sin a is negative, and therefore,
Example 3. Calculate tan α if cos α = √5/5 and α ∈ (π, 3π/2)
For any angle a belonging to the indicated interval tanα is positive and cos a is negative, and therefore
➢ Factorisation of the Sum Or Difference of Two sines Or cosines:
➢ Transformation of Products Into Sum Or Difference Of sines & cosines:
Example 4. Suppose x and y are real numbers such that tan x + tan y = 42 and cot x + cot y = 49. Find the value of tan(x + y).
tan x + tan y = 42 and cot x + cot y = 49
now, cot x + cot y = 49
Example 5. If x sinθ = y sin(θ + 2π/3) = z sin (θ + 4π/3) then:
(a) x + y + z = 0
(b) xy + yz + zx = 0
(c) xyz + x + y + z = 1
(d) none
Correct Answer is Option (b)
= xy + yz + zx = 0
Example 6. Find θ satisfying the equation, tan 15° . tan 25° . tan 35° = tan 0, where θ ∈ (0, 15°).
LHS = tan 15° . tan (30° - 5°) . tan (30° + 5°)
let t = tan 30° and m = tan 5°
Example 7. If tan A & tan B are the roots of the quadratic equation, a x2 + b x + c = 0 then evaluate a sin2 (A + B) + b sin (A + B) . cos (A + B) + c cos2 (A + B).
Now E = cos2 (A + B) [a tan2 (A + B) + b tan (A + B) + c]
Example 8. Show that cos2A + cos2(A + B) + 2 cosA cos(180° + B) · cos(360° + A + B) is independent of A. Hence find its value when B = 810°.
cos2A + cos2(A + B) - [2 cosA · cosB · cos (A + B)]
= cos2A + cos2(A + B) - [ {cos(A + B) + cos(A - B) } cos (A + B) ]
= cos2A + cos2(A + B) - cos2(A + B) - (cos2A - sin2B)
= sin2B which is independent of A
now, sin2(810°) = sin2(720° + 90°) = sin290° = 1
If A+B+C = π
Example 10. If A + B + C = π, prove that,
Solution:
Example 11. If A + B + C = θ and cotθ = cot A + cot B + cot C, show that , sin(A-θ). sin (B-θ).sin (C-θ)= sin3 θ.
Solution:
Given cot θ = cot A + cot B + cot C or cot θ - cot A = cot B + cot C
....(1)
....(2)
....(3)
Multiplying (1) , (2) and (3) we get the result
Example 12. Find whether a triangle ABC can exists with the tangents of its interior angle satisfying, tan A = x, tan B = x + 1 and tan C = 1 - x for some real value of x. Justify your assertion with adequate reasoning.
Solution:
In a triangle ∑ tan A = π tan A (to be proved)
x + x + 1 + 1 – x = x(1 + x)(1 – x)
2 + x = x – x3; x3 = –2, x = -21/3
Hence tanA = x < 0 and tanB = x + 1 = 1 – 21/3 < 0
Hence A and B both are obtuse. Which is not possible in a triangle. Hence no such triangle can exist.
Example 13. Find the greatest value of c such that system of equations x2 + y2 = 25; x + y = c has a real solution.
Solution:
Put x = 5 cosθ and y = 5 sinθ
5(cosθ+ sinθ) = c; but (cosθ + sinθ)max = √2 and (cosθ + sinθ)min = – √2
hence, cmax = 5√2
Example 14. Find the minimum and maximum value of f (x, y) = 7x2 + 4xy + 3y2 subjected to x2 + y2 = 1.
Solution:
Let x = cosθ and y = sinθ
y = f (θ) = 7 cos2θ + 4 sin θcosθ + 3 sin2θ = 3 + 2 sin 2q + 2(1 + cos 2θ)
= 5 + 2(sin 2θ + cos 2θ) but -√2 ≤ (sin 2θ + cos 2θ) ≤ √2
ymax = 5 + 2√2 and ymin = 5 – 2√2
1. What are the basic trigonometric identities? | ![]() |
2. What are the important trigonometric ratios? | ![]() |
3. What are the trigonometric functions of sum or difference of two angles? | ![]() |
4. What are the maximum and minimum values of trigonometric functions? | ![]() |
5. What are the conditional identities in trigonometry? | ![]() |
156 videos|176 docs|132 tests
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156 videos|176 docs|132 tests
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