Question

Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) cos(θ) − sin(θ) = 1

Answer 1

θ = {0, 3π/2} +2kπ

Explanation:

For solving problems involving a linear combination of sine and cosine, it can be helpful to use the identity ...

`asin(\theta)+bcos(\theta)=csin((\theta+\phi)) \quad\text{where $c=√(a^2+b^2)$ and $\phi=\tan^(-1)(b/a)$}`

Filling in the numbers, we have ...

`a=-1,b=1,c=√(2),\phi=\tan^(-1){(-1)}=(3\pi)/(4)\\\\√(2)\sin{(\theta+(3\pi)/(4))}=1\\\\\text{Solve for $\theta$:}\\\\\theta+(3\pi)/(4)=\sin^(-1){((1)/(√(2)))}\\\\\theta=\{(\pi)/(4),(3\pi)/(4)\}-(3\pi)/(4)+2k\pi\\\\\theta=\{0,(3\pi)/(2)\}+2k\pi\\\\\theta\approx \{0,4.712\}+6.283k`