📏 trigonometry

1. **Problem statement:** From a viewing tower 30 m above the ground, the angle of depression to an object on the ground is 36°. The angle of elevation to an aircraft vertically ab

1. The problem asks for the definitions of the tangent and cosine functions in trigonometry. 2. The tangent of an angle in a right triangle is defined as the ratio of the length of

1. **State the problem:** Find $\sin(B)$ for a right triangle where the vertical side is $\sqrt{13}$, the horizontal side is 6, and the hypotenuse is 7. Angle $B$ is between the ve

1. The problem is to prove the identity $$\arctan(1) + \arctan(2) + \arctan(3) = \pi$$. 2. We use the formula for the sum of arctangents:

1. The problem asks to find the values of $\sin 135^\circ$ and $\tan 15^\circ$.\n\n2. Recall the formulas and identities:\n- $\sin(180^\circ - \theta) = \sin \theta$\n- $\tan(45^\c

1. **نص المسألة:** حل المعادلة المثلثية $$2 \sin 2\theta + \sin \theta = 1$$ حيث \(\theta\) بالدرجات. 2. **القوانين المستخدمة:**

1. **Problem statement:** We need to find the height of a tree given two points A and B on the ground, 30 m apart, with a right angle (90°) between them at the tree's base. The ang

1. The problem asks to find the angle $\theta$ in radians such that $\sin \theta = \frac{\sqrt{2}}{2}$. 2. Recall that on the unit circle, the sine of an angle $\theta$ corresponds

1. The problem is to find the value of $\sin 240^\circ$ using the unit circle. 2. Recall that the unit circle defines sine as the y-coordinate of the point on the circle at a given

1. **State the problem:** We are given a table of $x$ and $y$ values and need to find the equation of the periodic function that fits these points. 2. **Given table:**

1. The problem asks to write the equation of the periodic function based on the graph. 2. The graph shows a cosine wave with amplitude 4, oscillating between 4 and -4, and a period

1. **State the problem:** We need to find the equation of the periodic function based on the given graph. 2. **Identify the form:** The function is given as $f(t) = a \sin(bt)$ or

1. The problem asks to compare the amplitudes of the functions $f(x) = -1.8 \cos x$ and $g(x) = -3.6 \cos x$. 2. Recall that the amplitude of a cosine function $a \cos x$ is the ab

1. **State the problem:** We need to find what fraction of a semicircle the angle 315° represents and then convert 315° to radians. 2. **Recall the facts:**

1. **State the problem:** Given $\sin x = \frac{p - q}{p + q}$ where $x$ is between $0^\circ$ and $90^\circ$, find $1 - \tan^2 x$. 2. **Recall the identity:** We know that $1 - \ta

1. **Problem statement:** Prove that $\tan 20^\circ \times \tan 40^\circ \times \tan 80^\circ = \sqrt{3}$. 2. **Recall the tangent triple-angle identity:**

1. مسئله: اگر \( \sin \alpha = -\frac{1}{5} \) و \( \alpha \) زاویه‌ای در ربع سوم باشد، مقدار \( \sqrt{25 - \cot^2 \alpha} \) را بیابید. 2. فرمول‌ها و نکات مهم:

1. **State the problem:** Solve the trigonometric equation $$\cos^2 x + 2 \sin x + \sin^2 x = 0$$. 2. **Use the Pythagorean identity:** Recall that $$\cos^2 x + \sin^2 x = 1$$.

1. The problem is to find the value of $\cot(-300^\circ)$. 2. Recall that the cotangent function is the reciprocal of the tangent function:

1. **Problem statement:** Prove that $$\sin^{-1} \frac{3}{5} + \frac{1}{2} \cos^{-1} \frac{5}{13} - \cot^{-1} 2 = \tan^{-1} \frac{28}{29}$$. 2. **Recall formulas and identities:**

1. **State the problem:** Solve the equation $$\sin \theta + 2 \sin \theta \cos \theta = 0$$ for $$\theta$$ in the range $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Write the equa