Advertisements
Advertisements
प्रश्न
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant
Advertisements
उत्तर
We know that sec2θ – tan2θ = 1
sec2θ – (– 2)2 = 1
sec2θ – 4 = 1
sec2θ = 1 + 4 = 5
sec θ = `+- sqrt(5)`
Since θ lies in the second quadrant sec θ is negative.
∴ sec θ = `- sqrt(5)`
cos θ = `1/sectheta = -1/sqrt(5)`
We know cos2θ + sin2θ = 1
`(- 1/sqrt(5))^2 + sin^2theta` = 1
`1/5 + sin^2theta` = 1
sin2θ = `1 - 1/5 = (5 - 1)/5`
sin2θ = `4/5`
sin θ = `+- 2/sqrt(5)`
Since θ lies in the second quadrant sin θ is positivee.
∴ sin θ = `2/sqrt(5)`
sin θ = `2/sqrt(5)`, cosec = `1/sintheta = sqrt(5)/2`
cos θ = `- 1/sqrt(5)`, sec θ = `1/costheta = - sqrt(5)`
tan θ = – 2, cot θ = `1/tantheta = - 1/2`
APPEARS IN
संबंधित प्रश्न
Find the values of `sin (-(11pi)/3)`
Find the value of the trigonometric functions for the following:
cos θ = `2/3`, θ lies in the I quadrant
Prove that `(cot(180^circ + theta) sin(90^circ - theta) cos(- theta))/(sin(270^circ + theta) tan(- theta) "cosec"(360^circ + theta))` = cos2θ cotθ
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of tan(x + y)
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Find the value of sin105°.
Prove that sin(π + θ) = − sin θ.
Prove that sin 105° + cos 105° = cos 45°
If cos(α – β) + cos(β – γ) + cos(γ – α) = `- 3/2`, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ = 0
Prove that cot(A + B) = `(cot "A" cot "B" - 1)/(cot "A" + cot "B")`
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`
Find the value of cos 2A, A lies in the first quadrant, when sin A = `4/5`
If θ is an acute angle, then find `cos (pi/4 + theta/2)`, when sin θ = `8/9`
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
Prove that (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ..... (1 + tan 44°) is a multiple of 4
Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x
Prove that `(sin x + sin 3x + sin 5x + sin 7x)/(cos x + cos x + cos 5x cos 7x)` = tan 4x
If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C
If A + B + C = `pi/2`, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C
