Determine the general solution of the following equation: 6 cos²x - 7 cos x - 3 = 0

Question 1
Given $\sin x = \frac{3}{5}$, find $\cos x$.

Step 1: Start with the Pythagorean identity.

Step 2: Substitute the given value.

Step 3: Isolate $\cos^2 x$.

Step 4: Take the square root (principal value for acute angle).

Assuming $x$ is acute, $\cos x = \frac{4}{5}$.

$\cos x = \dfrac{4}{5}$

Question 2
Given $\cos x = \frac{4}{5}$, find $\sin x$.

Step 1: Start with the Pythagorean identity.

Step 2: Substitute the given value.

Step 3: Isolate $\sin^2 x$.

Step 4: Take the square root (principal value for acute angle).

Assuming $x$ is acute, $\sin x = \frac{3}{5}$.

$\sin x = \dfrac{3}{5}$

Question 4.1
Simplify $\sin(90^\circ - \theta)$.

Step 1: In a right triangle, $\sin(90^\circ - \theta)$ is opposite over hypotenuse for angle $(90^\circ - \theta)$.
The opposite side to $(90^\circ - \theta)$ is the adjacent side to $\theta$.

Step 2: Thus, $\sin(90^\circ - \theta) = \frac{adjacent to \theta}{hypotenuse} = \cos \theta$.

$\sin(90^\circ - \theta) = \cos \theta$

Question 4.2
Simplify $\cos(90^\circ - \theta)$.

Step 1: $\cos(90^\circ - \theta)$ is adjacent over hypotenuse for angle $(90^\circ - \theta)$.
The adjacent side to $(90^\circ - \theta)$ is the opposite side to $\theta$.

Step 2: Thus, $\cos(90^\circ - \theta) = \frac{opposite to \theta}{hypotenuse} = \sin \theta$.

$\cos(90^\circ - \theta) = \sin \theta$

Question 4.3
Simplify $\tan(90^\circ - \theta)$.

Step 1: $\tan(90^\circ - \theta) = \frac{opposite to (90^\circ - \theta)}{adjacent to (90^\circ - \theta)}$.

Step 2: Opposite to $(90^\circ - \theta)$ is adjacent to $\theta$, adjacent to $(90^\circ - \theta)$ is opposite to $\theta$.

Step 3: Thus, $\tan(90^\circ - \theta) = \frac{adjacent to \theta}{opposite to \theta} = \cot \theta$.

$\tan(90^\circ - \theta) = \cot \theta$

Question 4.4
Prove $\sin^2 \theta + \cos^2 \theta = 1$ using the right triangle.

Step 1: Label the right triangle with hypotenuse $r$, opposite side to $\theta$ is $y$, adjacent side is $x$.

Step 2: By Pythagorean theorem in the triangle,

Step 3: Divide both sides by $r^2$.

$\sin^2 \theta + \cos^2 \theta = 1$