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Grade 10Trigonometry

If it is ggiven that sin²x + 3cosx-2 = 0, then find the value of cos³x + sec³x

Shashank Sheedhar Bhatt

8 Years agoGrade 10

2 Answers

Arun

8 Years ago

Dear Student

sin^2x = 1 – cos^2 x

now

1 – cos^2 x + 3 cos x – 2=0

cos^2 x- 3cos x +1 = 0

let cos x = t

then it will form a quadratic equation

t^2 – 3t + 1 = 0

t = (3 + $\sqrt$ 5)/2 and t = (3 – $\sqrt$ 5)/2

1^st value is cancelled

now

cos x= (3 – $\sqrt$ 5)/2

now you can easily find out the answer.

regards

Arun (askIITians forum expert)

Rishabh Sankla

6 Years ago

Dear Student,

Hope this will help you.

$sin^2 x + 3 cos x = 2$

$\Rightarrow (1-cos^2 x) + 3 cos x = 2$

$\Rightarrow cos^2 x - 3 cos x = -1$ [After shifting terms]

$\Rightarrow cos x(cos x - 3) = -1$

$\Rightarrow cos x - 3 = -sec x$

$\Rightarrow cos x + sec x = 3$

$\Rightarrow (cos x + sec x)^3 = (3)^3$

$\Rightarrow cos^3 x + sec^3 x + 3 cos x sec x(cos x + sec x) = 27$

$\Rightarrow cos^3 x + sec^3 x + 3\times3 = 27$

$\Rightarrow cos^3 x + sec^3 x = 18$

This is the simpler method to find answer to this question.

Regards

Rishabh

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