📏 trigonometry

1. **State the problem:** Simplify the expression $$\sin\left(\frac{5\pi}{4}\right) \cos\left(\frac{3\pi}{4}\right) + \sin\left(\frac{5\pi}{3}\right) \cos\left(\frac{13\pi}{6}\righ

1. Plantegem el problema: Tenim que $\cos(\alpha) = -\frac{3}{4}$ i $\tan(\alpha) > 0$. Cal calcular el valor de l'expressió $$\frac{1}{\csc(\alpha)} + \sec^2(\alpha).$$ 2. Recorde

1. We start with the problem: simplify or express \(\sin 2x\) in terms of \(\sin x\) and \(\cos x\). 2. The double-angle formula for sine is:

1. Muammo: $29(0-3-53). \cos \alpha = \frac{1}{2} (\sqrt{2} + \sqrt{3})$ tenglik uchun $\alpha$ ning o'tkir burchak qiymatini toping. 2. Formulalar va qoidalar:

1. Énonçons le problème : Montrer que $$\sin(2a) = \frac{2\tan a}{1 + \tan^2 a}$$. 2. Rappelons la formule trigonométrique de l'angle double pour le sinus :

1. **Énoncé du problème** : Pourquoi la phrase "Si le point M(x ; y ) \in G_{\cos} alors N(x + \pi ; y ) \in G_{\cos}" est-elle fausse ?

1. Muammo: cos\alpha = \frac{1}{2} \sqrt{2} + \sqrt{3} tenglik qaysi \alpha o‘tkir burchak uchun to‘g‘ri?\n\n2. Formulalar va qoidalar: cos\alpha qiymati -1 dan 1 gacha bo‘ladi. \n

1. Masala: Agar $0 < \alpha < \frac{\pi}{2}$ va $\cos \alpha = \frac{1}{2} \sqrt{2} + \sqrt{2}$ bo'lsa, $\alpha$ ning qiymatini toping. 2. Formulalar va qoidalar: $\cos \alpha$ qiy

1. Masala: $0 < \alpha < \frac{\pi}{2}$ va $\cos \alpha = \frac{1}{2} \sqrt{2} + \sqrt{2}$ bo'lsa, $\alpha$ ning qiymatini toping. 2. Masala: $\frac{\sin^4 \alpha + 2 \cos \alpha \

1. **State the problem:** We need to find the distance between the boat and the top of the lighthouse. The lighthouse height is 31.7 m, and the horizontal distance from the boat to

1. We are asked to calculate the tangent and cotangent of angle $\alpha$ given that $\sin \alpha = \frac{1}{6}$ and that $\alpha$ lies in the first quadrant. 2. Recall the definiti

1. **State the problem:** We are given the function $y = 3 \sin\left(\theta - \frac{\pi}{2}\right)$ and want to understand its properties and graph. 2. **Recall the sine function p

1. **State the problem:** We have a right triangle with a right angle at vertex P.

1. **State the problem:** We have a right triangle with a right angle at vertex T, hypotenuse 95, angle at W is 31°, and side opposite angle W is $x$. We need to find $x$. 2. **For

1. **State the problem:** We need to find the length of side $x$ (SU) in right triangle SUT where angle $S = 72^\circ$, side $TU = 9.4$, and angle $T$ is the right angle. 2. **Iden

1. We are asked to prove the identity: $$\frac{\tan \alpha - \tan \beta}{1 - \tan \alpha \tan \beta} = \frac{\tan \alpha + \tan \beta}{1 + \tan \alpha \tan \beta}$$

1. The problem is to understand and simplify the expression $\sin(x) + \cos$. 2. Here, $\sin(x)$ is the sine function of variable $x$, which is well-defined. However, $\cos$ alone

1. **State the problem:** We are given a sine function of the form $f(x) = A \sin(Bx - C)$ and a graph with specific points: amplitude 1, zero at $x = -\frac{\pi}{6}$, peak at $x =

1. El problema es encontrar el valor de $\sin^{-1}\left(\frac{1}{2}\right)$ y $\tan^{-1}(-1)$, y además indicar el dominio y rango de cada función. 2. Recordemos que $\sin^{-1}(x)$

1. مسئله: محاسبه \( \sin\left(\frac{\pi}{8}\right) \) و \( \sin\left(\frac{3\pi}{8}\right) \) و سپس یافتن تقریب عددی مشتق \( \cos\left(\frac{3\pi}{8}\right) \) با استفاده از مشتق‌گ

1. Let's start by understanding what trigonometry is. Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially