📏 trigonometry

1. **Problem statement:** Jack and Sangita stand on opposite sides of a 10-m flagpole. Jack sees the top of the pole at an angle of elevation of 50°, and Sangita sees it at 35°. We

1. **State the problem:** Prove the identity $$\frac{\sin 2x}{1 + \cos 2x} \cdot \frac{\cos x}{1 + \cos x} = \tan \frac{x}{2}$$. 2. **Recall key formulas:**

1. **Problem:** Solve triangle \(\triangle ABC\) given \(A=36^\circ\) and \(c=9\). We need to find sides \(a, b\) and angles \(B, C\).\n\n2. **Formula:** Use Law of Sines: $$\frac{

1. مسئله: معادله $$0 = 4 - x - x^2 - x^3$$ داده شده است و $$\tan \alpha$$ و $$\tan \beta$$ جواب‌های این معادله هستند. باید مقدار $$\tan(\alpha + \beta)$$ را پیدا کنیم. 2. ابتدا معا

1. **Problem statement:** Kristoffer's kite string makes an angle of 15° with the ground, and the string length is $\sqrt{6} + \sqrt{2}$ meters. We need to find how high the kite i

1. The problem asks to find the value of $\csc(\csc^{-1}(7))$. 2. Recall that $\csc^{-1}(x)$ is the inverse cosecant function, which returns an angle $\theta$ such that $\csc(\thet

1. مسئله را بیان می‌کنیم: \[ \frac{1}{4} = \frac{\sin^{7} \alpha + 1}{1 + \cot^{7} \alpha} - \frac{\cos^{7} \alpha}{1 + \tan^{7} \alpha} \]

1. مسئله: مساحت مثلث ABH برابر با $$\frac{1}{4} \sin \alpha$$ است و باید مقدار $$\tan \alpha$$ را پیدا کنیم. 2. فرمول مساحت مثلث با قاعده و ارتفاع: $$\text{مساحت} = \frac{1}{2} \ti

1. مسئله: اگر $0 < \alpha < 45$ باشد، مقدار عبارت $$\frac{\tan \alpha -1}{\cot \alpha -1}$$

1. مسئله: مقدار $\tan \alpha$ را پیدا کنید اگر $\cos \alpha = \frac{1}{3}$ و زاویه $\alpha$ در ناحیه چهارم قرار دارد. 2. فرمول‌ها و نکات مهم:

1. مسئله: مقدار $\cos \alpha$ را برای نقطه $P(x, \frac{2}{3} \pi)$ در دایره مثلثاتی پیدا کنید. 2. در دایره مثلثاتی، مختصات نقطه روی دایره به صورت $(\cos \alpha, \sin \alpha)$ است ک

1. مسئله: اگر $\sin \theta$ و $\cot \theta$ هم‌علامت باشند، باید تعیین کنیم که زاویه $\theta$ در کدام ربع مثلثاتی قرار دارد. 2. فرمول‌ها و نکات مهم:

1. The problem asks: In the curve $y = \tan 4x$, what is its period? 2. The general formula for the period of $y = \tan bx$ is $\frac{\pi}{|b|}$.

1. **State the problem:** We have a right triangle GHI with a right angle at H. Side GH is 5.6 units, angle at I is 29°, and GI is the hypotenuse labeled $x$. We need to find $x$.

1. **Problem statement:** Simplify the expressions involving inverse trigonometric functions: (i) $\arcsin(\sin(\frac{2\pi}{5}))$, $\arccos(\cos(\frac{2\pi}{5}))$, $\arctan(\tan(\f

1. **Problem statement:** Simplify the following expressions involving inverse trigonometric functions: (i) $\arcsin(\sin(\frac{2\pi}{5}))$, $\arccos(\cos(\frac{2\pi}{5}))$, $\arct

1. **Problem statement:** Given a point $P(0) = \left(-\frac{1}{2}, -\frac{1}{\sqrt{2}}\right)$ on the unit circle corresponding to angle $\theta$, find the coordinates of $P(\thet

1. **State the problem:** Solve the equation $5\cos^2 x + 4\cos x = 1$ for $x$. 2. **Rewrite the equation:** Let $y = \cos x$. The equation becomes:

1. **State the problem:** Solve the trigonometric equation $$5 \cos^2 x + 4 \cos x = 1$$ for $$0 \leq x \leq 2\pi$$. 2. **Rewrite the equation:** Move all terms to one side to set

1. The problem asks to find the endpoint of the radius of the unit circle corresponding to 120 degrees. 2. Recall that the unit circle has radius 1, and the coordinates of a point

1. **State the problem:** Calculate the following trigonometric values and verify identities: - $\cot 30^\circ$