Advertisements
Advertisements
Question
If sinθ + cosθ = 1, then find the general value of θ.
Solution
Given, sinθ + cosθ = 1
On dividing both the sides by `sqrt2`,
`sintheta/sqrt2 + costheta/sqrt2 = 1/sqrt2`
⇒ `cos(theta - pi/4) = cos pi/4`
⇒ `theta - pi/4 = pi/4`
We know that, θ = 2nπ ± α when cosθ = cosα
⇒ `theta - pi/4 = 2npi ± pi/4, n ∈ z`
⇒ `theta = 2npi ± pi/4 + pi/4`
Taking the positive sign,
⇒ `theta = 2npi + pi/4 + pi/4`
⇒ `theta = 2npi + pi/2`
Taking the Negative sign,
⇒ `theta = 2npi - pi/4 + pi/4`
⇒ θ = 2nπ, n ∈ z
So, the general value is `theta = 2npi + pi/2` and θ = 2nπ.
APPEARS IN
RELATED QUESTIONS
Find the value of: sin 75°
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Prove the following:
`(sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
sin (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)
Prove that
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
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:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
If sin α + sin β = a and cos α + cos β = b, show that
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Show that sin 100° − sin 10° is positive.
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If A + B = C, then write the value of tan A tan B tan C.
If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
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 tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
