📏 trigonometry

1. Let's clarify the problem: You want to solve a trigonometric equation for $x$ with conditions like $0 \leq x \leq 360$ degrees. 2. Typically, such problems involve equations lik

1. **State the problem:** Evaluate the expression $$\frac{2 \cos \left(\frac{8\pi}{6}\right) - 5 \sin \left(-\frac{5\pi}{2}\right)}{3 \tan \left(\frac{3\pi}{4}\right)}$$. 2. **Simp

1. Let's start by stating the problem: solving a trigonometric equation means finding all angles $x$ that satisfy an equation involving trigonometric functions like $\sin x$, $\cos

1. **State the problem:** Prove that $$\sqrt{2} + \sqrt{2} + \cos 4\theta = 2 \cos \theta$$. 2. **Simplify the left side:** Note that $$\sqrt{2} + \sqrt{2} = 2\sqrt{2}$$, so the ex

1. The problem is to solve the equation $A \sin(x) + B \cos(x) = 0$ for $x$. 2. We start by isolating one of the trigonometric functions. Rewrite the equation as:

1. **State the problem:** We have a right triangle with an angle of 68° and the side opposite this angle is 16 m. We need to find the length of the adjacent side $x$. 2. **Identify

1. **State the problem:** Given the equation $$8 + \csc^2 \theta = 6 \cot \theta,$$ find the value of $$\tan \theta$$. 2. **Recall trigonometric identities:**

1. The problem gives us an angle $\frac{5\pi}{12}$ radians and a sine value $\sin \theta = \frac{5}{7}$ with $0 < \theta < 90^\circ$. We need to understand the relationship or solv

1. Express the angle in radian measure as a multiple of $\pi$ radians. Recall that $180^\circ = \pi$ radians.

1. **Convert degrees to radians using the formula:** $$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$$

1. Statement of the problem. Problem: Points A, B, C lie on level ground with C due east of B, $\angle ABC=80^\circ$ and $\angle ACB=20^\circ$. Calculate the bearing of (a) C from

1. **State the problem:** Prove the expression $$\frac{2\tan x - \sin 2x}{2\sin^2 x}$$ simplifies or equals a certain value. 2. **Rewrite the expression:** Recall that $$\tan x = \

1. Convert from degrees to radians. (a) Convert 300° to radians.

1. State the problem: Simplify $\frac{\cos(\theta)}{1-\sin(\theta)} - \tan(\theta)$.\n2. Multiply the first fraction by $\frac{1+\sin(\theta)}{1+\sin(\theta)}$ to rationalize the d

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent term:** Recall that $$\tan(\theta) = \frac{\sin

1. The unit circle is a circle with radius 1 centered at the origin (0,0) on the coordinate plane. 2. The coordinates of any point on the unit circle can be described using an angl

1. **State the problem:** (a) Prove the double angle formulas:

1. We start by proving the double-angle formulae: (a)(i) Prove that $\cos 2A = 2 \cos^2 A - 1$.

1. **Problem 1:** Given $\tan \theta = \frac{a}{b}$, find $$\frac{a \sin \theta + b \cos \theta}{a \sin \theta - b \cos \theta}.$$ Since $\tan \theta = \frac{a}{b}$, we have $\sin

1. We start with the problem: Prove that $\sec^{2}\theta - \cos^{2}\theta = 1$. 2. Recall that $\sec\theta = \frac{1}{\cos\theta}$ by definition.

1. **State the problem:** Calculate the value of $$\cos 60^\circ \cos 30^\circ + \sin 60^\circ + \sin 30^\circ$$. 2. **Recall values of trigonometric functions:**