📏 trigonometry

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:**

1. The problem is to simplify the expression $\cos 60\cos 30 + \sin 60\sin 30$.\n\n2. Recognize that this expression matches the cosine addition formula: $$\cos(a - b) = \cos a \co

1. Дано рівняння: $$2\sin^4 x - 2\cos^4 x - 1 = 0$$ у проміжку $$[-\pi, \pi]$$. 2. Використаємо формулу різниці квадратів: $$a^2 - b^2 = (a-b)(a+b)$$.

1. We start with the given equation: $$\tan 3A \cdot \tan 2A \cdot \tan A = \tan 3A + \tan 2A + \tan A$$ 2. The goal is to verify or simplify this relationship.

1. **State the problem:** We have a right-angled triangle with hypotenuse length 14 cm, an adjacent side to angle $x$ of length 7 cm, and angle $x$ to find. 2. **Identify the trigo

1. **State the problem:** We have a right-angled triangle with a 60° angle, hypotenuse of length 16 cm, and side adjacent to the 60° angle labeled as $x$. We need to find the exact

1. Stating the problem: Evaluate the expressions (h) $$\frac{\sin 30^\circ}{\tan 45^\circ} + \frac{\sin 90^\circ}{3}$$

1. Stating the problem: Evaluate the expressions (b) \quad \frac{\sin 30^\circ}{\cos 30^\circ} - \tan 30^\circ

1. The problem is to evaluate the functions: $$f(1) = \cos 0^\circ \times \sin 90^\circ$$