📏 trigonometry

1. **Problem statement:** Solve the equation $$\cos 3x + \cos 2x + \cos x = 0$$ for $$0 \leq x \leq 2\pi$$. 2. **Formula and identities:** Recall the cosine addition formulas and s

1. Le problème est de comprendre pourquoi $\arctan(-1,742) = -1,05$ radians. 2. La fonction $\arctan(x)$ donne l'angle dont la tangente est $x$. Elle est définie pour tout réel $x$

1. The problem is to evaluate $\csc^2(45^\circ) - \tan^2(45^\circ)$.\n\n2. Recall the trigonometric identities and values:\n- $\csc(\theta) = \frac{1}{\sin(\theta)}$\n- $\tan(\thet

1. **Problem Statement:** We need to find the length of side BC in a right triangle where angle A is 35°, side AC (adjacent to angle A) is 2 units, and the right angle is at vertex

1. **State the problem:** We need to find the height of a balloon observed from two stations X and Y, which are 3000 feet apart. Given angles are horizontal angles and angle of ele

1. **Problem statement:** A balloon is observed from two stations, X and Y, which are 300 meters apart. At station X, the horizontal angle between the balloon and point C is $75^\c

1. **Problem:** Given a right triangle \(\triangle ACB\) with side \(b=12\) and angle \(\angle A=12^\circ\), find the missing parts (sides \(a\), \(c\) and angle \(\angle B\)). 2.

1. The problem is to understand why $\cos \frac{\pi}{2} = 0$. 2. Recall that the cosine function relates to the unit circle, where $\cos \theta$ is the x-coordinate of the point on

1. مسئله: بررسی اینکه آیا نقطه $\left(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}\right)$ روی نیم‌خط $OP$ در ربع مثبت محور $OX$ قرار دارد یا خیر. 2. فرمول و توضیح: نیم‌خط $OP$ معمولاً ب

1. **State the problem:** Simplify the expression $$\frac{2+\tan^2 x}{\sec^2 x} - 1 = g(x)$$. 2. **Recall the identity:** We know that $$\sec^2 x = 1 + \tan^2 x$$.

1. **State the problem:** We need to find the horizontal distance from the boat to the foot of the cliff. The cliff height is 55 m, and the angle of depression from the top of the

1. **Énoncé du problème :** Nous avons un triangle avec les côtés $A=5,9$, $B=3,4$ et l'angle $\alpha = 22^\circ$ opposé au côté $a$. Nous devons trouver l'angle $c$ (noté ici $\ga

1. **State the problem:** Evaluate the expression $\cos^2(10^\circ) + \cos^2(50^\circ) - \sin(40^\circ) \sin(80^\circ)$.\n\n2. **Recall relevant formulas:** Use the Pythagorean ide

1. **State the problem:** Given acute angles $A$ and $B$ such that $\sin(A-B)=0$ and $2\cos(A+B)-1=0$, find the values of $A$ and $B$. 2. **Use the given equations:**

1. The problem asks for the interval of the principal value of the function $\cos^{-1} x$, also known as the inverse cosine or arccosine function, and to draw its graph. 2. The pri

1. **Problem statement:** We have a field ABCD with given sides and angles. We need to find (a) the length of CD and (b) the angle ABD. 2. **Given data:**

1. The problem asks for the interval of the principal value of the function $\cos^{-1} x$ (arccosine of $x$) and to draw its graph. 2. The principal value of the inverse cosine fun

1. **State the problem:** We need to find the equation of a sinusoidal function based on the given graph description. 2. **Identify key features from the graph:**

1. The problem is to convert the expression $\frac{2\pi^c}{9}$ into degrees. 2. Recall that $\pi$ radians equals 180 degrees. To convert radians to degrees, multiply by $\frac{180}

1. **State the problem:** Convert the expression $\frac{2\pi^c}{9}$ into degrees. 2. **Recall the conversion formula:** To convert radians to degrees, use the formula:

1. **State the problem:** We have a ladder leaning against a wall forming a right-angled triangle. The ladder is the hypotenuse of length 5.2 m, and the base (distance from the wal