Advertisements
Advertisements
प्रश्न
Show that sin 100° − sin 10° is positive.
Advertisements
उत्तर
\[\text{ Let } f\left( \theta \right) = \sin100° - \sin10°\]
\[\text{ Multiplying and dividing by } \sqrt{1^2 + 1^2}, i . e . \text{ by } \sqrt{2}, \text{ we get }: \]
\[ \sqrt{2}\left( \frac{1}{\sqrt{2}}\sin100° - \frac{1}{\sqrt{2}}\sin10°\right)\]
\[ = \sqrt{2}\left( \cos45°\sin(90°+ 10°) - \sin45°\sin10°\right)\]
\[ = \sqrt{2}\left( \cos45°\cos10° - \sin45°\sin10° \right)\]
\[ = \sqrt{2}\cos(45°+ 10°) = \sqrt{2}co s55° , \text{ which is positive since \cos is positive in the first quadrant } .\]
APPEARS IN
संबंधित प्रश्न
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Find the value of: sin 75°
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
sin (A − B)
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
cos (A + B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that:
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]
Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
Prove that:
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Write the maximum value of 12 sin x − 9 sin2 x.
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.
If \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\] then write the value of tan x tan y.
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If A + B + C = π, then \[\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\] is equal to
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
If tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β is equal to ______.
The value of sin(45° + θ) - cos(45° - θ) is ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.
In the following match each item given under the column C1 to its correct answer given under the column C2:
| Column A | Column B |
| (a) sin(x + y) sin(x – y) | (i) cos2x – sin2y |
| (b) cos (x + y) cos (x – y) | (ii) `(1 - tan theta)/(1 + tan theta)` |
| (c) `cot(pi/4 + theta)` | (iii) `(1 + tan theta)/(1 - tan theta)` |
| (d) `tan(pi/4 + theta)` | (iv) sin2x – sin2y |
