Advertisements
Advertisements
Question
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant
Advertisements
Solution
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
RELATED QUESTIONS
Find the values of sin(480°)
Find the values of cot(660°)
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)
If sin A = `3/5` and cos B = `9/41 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of sin(A + B)
Prove that sin(π + θ) = − sin θ.
Find a quadratic equation whose roots are sin 15° and cos 15°
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
Find the value of cos 2A, A lies in the first quadrant, when sin A = `4/5`
If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2
Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ
Express the following as a sum or difference
sin 5θ sin 4θ
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
Show that cot(A + 15°) – tan(A – 15°) = `(4cos2"A")/(1 + 2 sin2"A")`
If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
