📏 trigonometry

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent in terms of sine and cosine:**

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. Rewrite $\tan(\theta)$ as $\frac{\sin(\theta)}{\cos(\theta)}$ to have

1. **State the problem:** We need to find the value of $\sin\left(\frac{4\pi}{6}\right)$.\n\n2. **Simplify the angle:** Simplify the fraction inside the sine function.\n$$\frac{4\p

1. **Problema**: Calcular os valores das funções trigonométricas dadas: (a) $\sin\left(\frac{4\pi}{6}\right)$ (b) $\cos\left(\frac{8\pi}{6}\right)$ (c) $\tan\left(\frac{5\pi}{4}\ri

1. The problem is to understand the expression $\cos x \sin x$. 2. This is a product of the cosine and sine trigonometric functions.

1. **State the problem:** Simplify the trigonometric expression $\sin x \cos^2 x - \sin x$. 2. **Factor the expression:** Notice that $\sin x$ is common in both terms, so factor it

1. For the triangle with the hypotenuse labeled "hyp" (longest side), the side opposite the acute angle is labeled "opp" and the side adjacent to the acute angle is labeled "adj".

1. **Problem statement:** We want to prove that $$\cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \cos^4\left(\frac{5\pi}{8}\right) + \cos^4\left(\frac{7\pi}

1. المشكلة المعطاة هي حل المعادلة التي عند حلها باستخدام الآلة يظهر الناتج كـ $\frac{\sqrt{3}}{3}$.\n\n2. نعلم أن $\frac{\sqrt{3}}{3}$ هو القيمة المُبسطة للعدد $\cot 60^{\circ}$ أو

1. Problem statement: Simplify $\frac{\cos\theta}{1-\sin\theta} - \tan\theta$. 2. Rewrite the tangent using $\tan\theta = \frac{\sin\theta}{\cos\theta}$.

1. The problem asks to verify if the given graph is correct for the function being a cosine wave. 2. The cosine function $y=\cos x$ starts from $y=1$ at $x=0$, decreases to $y=-1$

1. **Problem 1: Draw the graph of** $y = \cos \theta^\circ$ for $0^\circ \leq \theta^\circ \leq 180^\circ$. - At $\theta = 0^\circ$, $y = \cos 0^\circ = 1$ (maximum).

**Problème :** Dans le triangle rectangle PDJ, l'angle droit est en J, la longueur PD = 1,9 cm et l'angle JPD = 62°. Il faut calculer la longueur JD, arrondie au dixième.

1. **State the problem:** Solve the equation $3\tan x = \sqrt{3}$ for $x$. 2. **Isolate $\tan x$:** Divide both sides by 3:

1. **State the problem:** Solve the inequality $$2\sin^2 x + \sin x - 1 < 0$$ for $x$. 2. **Rewrite the inequality:** Let $u = \sin x$. The inequality becomes $$2u^2 + u - 1 < 0.$$

1. **Stating the problem:** Solve the inequality $$2\sin^2 x + \sin x - 1 < 0$$ for $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{6}\right)$$. 2. **Rewrite the inequality:** Let $$t = \

1. The problem is to find the correct values of $\cos 120^\circ$ and $\sin 120^\circ$ among the given options. 2. Recall that $120^\circ$ is in the second quadrant where cosine is

1. We start with the given expression: $$\frac{1}{\sin\theta} - \frac{\sin^2\theta}{\sin\theta}$$\n\n2. Since both terms have the same denominator $\sin\theta$, combine them under

1. **State the problem:** We have two trees 20 m apart horizontally. The first tree is 12 m tall. From the top of the first tree, the angle of elevation to the top of the second tr

1. **State the problem:** Two trees stand 20 m apart horizontally. The first tree is 12 m tall. From its top, the angle of elevation to the top of the second tree is 30°, and the a

1. **State the problem:** An aeroplane flying at 6000 m height passes vertically above another plane, and from an observer's point on the ground, the angles of elevation to the two