📏 trigonometry

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

1. **State the problem:** We need to find the values of $\cos(-3.45\pi)$ and $\sin(-3.45\pi)$ using the unit circle.

1. **Problem:** Find the measurements $m_a$, $m_b$, and $m_c$ given the relation $\frac{\sin A}{18} = \frac{\sin 54}{31}$. 2. **Step 1:** Use the Law of Sines formula:

1. The problem involves understanding the general form of a sine function: $$y = A \sin(B(x - C)) + D$$ where: - $A$ is the amplitude (height of the wave peaks).

1. **State the problem:** We have a right triangle ABC with a right angle at C, angle A = 65°, and hypotenuse AB = 21 units. We need to find the lengths of sides a (BC) and b (AC).

1. **State the problem:** Solve for the side $x$ in a right triangle where one leg is 30, the right angle is between sides 30 and $x$, and the angles are 65° opposite side 30 and 5

1. **Problem:** Find the angle measure $C$ given $\cos C = 0.9063$.\n\n2. **Formula:** Use the inverse cosine function to find the angle: $$C = \cos^{-1}(0.9063)$$\n\n3. **Calculat

1. **State the problem:** Prove the trigonometric identity $$\sin^2 \theta + \cos^2 \theta + \tan^2 \theta = \sec^2 \theta$$. 2. **Recall fundamental identities:**

1. **State the problem:** We have a periodic function $h(t)$ modeling the distance between point B on a rotating boat motor blade and the water line. The motor blades rotate 50 tim

1. **Problem statement:** We have a triangle with points A, B, C on a straight line and a right angle at B. Given angle $\angle A = 20^\circ$, length $AB = 12.6$ cm, and length $DC

1. **State the problem:** We are given the graph of the function $y = a \cos(bx) + c$.

1. **State the problem:** We have a right triangle with an angle of $53^\circ$ at vertex O, the side adjacent to this angle is 1.8, and the side opposite this angle is $x$. We need

1. **State the problem:** We have a right triangle POQ with a right angle at P, angle $\angle O = 53^\circ$, opposite side to $\angle O$ is $PQ = 1.8$, and adjacent side to $\angle

1. **State the problem:** We need to solve for $x$ in a right triangle $\triangle TYW$ where $\angle T$ is $90^\circ$, side $YT = x$, side $YW = 95$, and $\angle W = 31^\circ$. 2.

1. **State the problem:** We need to write an equation for the height $h = f(t)$ of a person on a ferris wheel as a function of time $t$ in minutes. 2. **Given information:**

1. **State the problem:** We need to write an equation for the height $h = f(t)$ of a person on a ferris wheel as a function of time $t$ in minutes. 2. **Given information:**

1. Let's understand the expression $\frac{x}{\sin x}$. 2. This is a ratio where the numerator is $x$ and the denominator is $\sin x$, the sine of $x$.

1. **Problem statement:** Two buildings Tai Chek Tower (height 200 m) and Seng Office Block are 25 m apart on level ground. The angle of depression from the top of Tai Chek Tower (