Solving Cos, Tan, and Sin Equations: Step-by-Step Guide | Homework Help

[pinnacleprouk]

Homework Statement

Please help solve these two equations (giving two solutions)

sin x =√ 2 cos x

sin^2 x - cos^2 x - 2sin x = 1


Any help no matter how little is much appreciated

Homework Equations

The Attempt at a Solution

Not sure how to solve these,

thanks in advance

 

[danago]

When solving these types of equations, one method is to try and use trigonometric identities to manupulate the equations so that only one type of trigonometric function is left.

For example, with the first one, what could you divide both sides by so that there is only sin, cos or tan left over?

 

[HallsofIvy]

Homework Statement

Please help solve these two equations (giving two solutions)

sin x =√ 2 cos x

$$\frac{sin x}{cos x}= tan x= \sqrt{2}$$
Can you solve that?

sin^2 x - cos^2 x - 2sin x = 1

Since $sin^2 x+ cos^2 x= 1$, $cos^2 x= 1- sin^2 x$ so this becomes
$$sin^2 x- (1- sin^2 x)- 2 sin x= sin^2 x- 1+ sin^2 x- 2sin x= 1$$
$$2sin^2 x- 2sin x- 2= 0$$
$$sin^2 x- sin x- 1= 0$$

Let y= sin x so the equation becomes $y^2- y- 1= 0$.

Any help no matter how little is much appreciated

Homework Equations

The Attempt at a Solution

Not sure how to solve these,

thanks in advance