📏 trigonometry

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

1. Stating the problem: We have two airplanes observed from the same point on the ground. The higher plane is 6000 m above the ground. The angles of elevation to the two planes are

1. Problem 129.a: Simplify $\sqrt{2} \sin\left(\frac{\pi}{4} + \alpha\right) - \sin \alpha$.\n Using the sine addition formula, $\sin(a+b) = \sin a \cos b + \cos a \sin b$, we get:

1. We are asked to find the value of $x$ given that the maximum value of $\prod_{i=1}^n \cos \alpha_i$ under the conditions $0 \leq \alpha_i \leq \frac{\pi}{2}$ and $\prod_{i=1}^n

1. We are asked to find the smallest positive integer $x$ such that $$\tan(x - 160) = \frac{\cos 50}{1 - \sin 50}.$$ 2. Notice the right side: it resembles the tangent half-angle i

1. The problem is to analyze the function $y = \cos x$ and understand its key features including intercepts and extrema. 2. The cosine function has the form $y = \cos x$, which is

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

1. Convert from degrees to radians. (a) To convert $300^\circ$ to radians, use the formula:

1. **State the problem:** Find the smallest positive integer $x$ (in degrees) such that $$\tan(x - 160^\circ) = \frac{\cos 50^\circ}{1 - \sin 50^\circ}.$$\