📏 trigonometry

1. The problem is to find the value of $\sin(3\theta)$ without using the triple angle formula. 2. We start with the angle addition formula for sine: $$\sin(a+b) = \sin a \cos b + \

1. **State the problem:** Simplify and solve the expression $$\frac{\cos(3A) + \sin(A)}{\cos(3A) - \cos(A)} + \tan A = -2 \cot(2A)$$ for angle $A$. 2. **Recall formulas and identit

1. **State the problem:** We need to find the tangent of angle $Q$ in right triangle $PQR$ where $\angle R$ is the right angle. 2. **Recall the definition of tangent:**

1. **State the problem:** We have a triangle with angles 45°, 60°, and 75° (since the sum of angles in a triangle is 180°). The side opposite the 45° angle is given as $18\sqrt{2}$

1. **Problem Statement:** Given a right triangle with angle $C = 90^\circ$, angle $B = 60^\circ$, side $AB = 12$, side $AC = b$, and side $BC = a$, find the relationship between th

1. **Problem statement:** We have a right triangle with angles 45°, 60°, and 90°. The side adjacent to the 45° angle is 2, the side opposite to 45° is $x$, the side opposite to 60°

1. **Problem Statement:** Find the values of \(\sin 120^\circ\) and \(\cos 225^\circ\) using the unit circle.

1. **Problem Statement:** Given a right triangle with legs 14 and 50, find the trigonometric ratios for angle R. 2. **Recall the definitions:**

1. **State the problem:** We have a right triangle with a 60° angle, the side opposite this angle is $6\sqrt{3}$, the adjacent side is $y$, and the hypotenuse is $x$. We need to fi

1. **Problem 12:** Given a right triangle with a 60° angle, side adjacent to 60° is $\frac{5\sqrt{3}}{3}$, side opposite 60° is $y$, and hypotenuse is $x$. Find $x$ and $y$. 2. **R

1. **State the problem:** We need to find the length $x$ in a triangle using the sine rule. Given angles are $40^\circ$ and $56^\circ$, and the side opposite the $56^\circ$ angle i

1. **State the problem:** We need to find the length of side AC in the triangle formed by the lighthouse, boat A, and boat B. 2. **Given information:**

1. The problem is to find the missing sides $x$ and $y$ in the right triangle with a given side 10 and an angle of 60° without using the Pythagorean theorem. 2. We use trigonometri

1. **Problem Statement:** Find the range and amplitude of the function $$y = \sin(x + 45)$$ and understand its graph shape. 2. **Recall the sine function properties:**

1. **Problem Statement:** Find the range, amplitude, and period of the function $$y = \cos x - 5$$ and understand its graph. 2. **Recall the cosine function properties:**

1. **State the problem:** Simplify the expression $$\sin(90^\circ - \alpha) + \cos(90^\circ + \alpha) - \sin(180^\circ + \alpha)$$. 2. **Recall trigonometric identities:**

1. Problem: Given point $P(x,y)$, find exact values of $\sin \beta$ and $\cos \beta$ where $\beta$ is the angle formed by the line from origin to $P$ with the positive x-axis. 2. F

1. **Problem statement:** We are given that Rory's shot length is 187 m, and we need to find the angle \( |\angle FTP| \) by which his shot is off target, rounded to the nearest de

1. مسئله اول: اگر $\sin x = \frac{4+m}{1+m}$ و $\frac{\pi}{4} < x - \frac{\pi}{4} < \frac{\pi}{4}$ باشد، مقدار $m$ را بیابید. 2. ابتدا بازه $x$ را ساده می‌کنیم:

1. مسئله را بیان می‌کنیم: معادله $\frac{9}{1} - m = \cos 2\alpha$ داده شده است و بازه $\frac{\pi}{4} < \alpha < \frac{3\pi}{4}$ است. باید حدود تغییرات $m$ را پیدا کنیم. 2. ابتدا مق

1. مسئله: مقدار $$\cos\left(\frac{3\pi}{2} - \alpha\right)$$ را پیدا کنید، با توجه به اینکه انتهای کمان $$\alpha$$ در ناحیه سوم مثلثاتی قرار دارد. 2. فرمول استفاده شده: از فرمول زا