📏 trigonometry

1. مسئله را بیان می‌کنیم: معادله $$\sin^{140^\circ}(a) + \cos^{140^\circ 5}(a) + \sin^{140^\circ 6}(a) = \frac{1}{3702}$$ را در بازه $$a \in \left[ \frac{\pi}{4} , \frac{3\pi}{4} \

1. **State the problem:** We have a camera positioned $x$ feet horizontally from the base of a missile launching pad. A missile of length $a$ feet is launched vertically. The base

1. **Giải phương trình a):** Bài toán: Giải phương trình $$\frac{2\cos 2x - 8 \cos x + 7}{\cos x} = 1$$.

1. **Prove** $ (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2 = 2 $. Step 1: Expand each square:

1. The problem states that $\cot \theta = \frac{\sqrt{3}}{2}$. We need to find $\theta$ or related trigonometric values. 2. Recall that $\cot \theta = \frac{\cos \theta}{\sin \thet

1. **State the problem:** Given that $\sec \theta = 3$, find the values of $\cos \theta$ and $\sin \theta$. 2. **Recall the definition:** $\sec \theta = \frac{1}{\cos \theta}$.

1. **State the problem:** We need to fill in the missing values in the table for the function $y = -\sec(x)$ at $x = 0.1$ and $x = 0.2$ to 3 decimal places. 2. **Recall the functio

1. The problem asks us to find the values of $y = \cos(x)$ for the given $x$ values: $-1$, $-\frac{2}{3}$, $-\frac{1}{3}$, $0$, $\frac{1}{3}$, $\frac{2}{3}$, and $1$. We will calcu

1. **State the problem:** We need to find the length $f$, which is the height of a right triangle opposite the $38^\circ$ angle. The base adjacent to this angle measures 7.2 cm. 2.

1. The problem is to determine if the function $\sin(x)$ is bijective. 2. A function is bijective if it is both injective (one-to-one) and surjective (onto).

1. The problem is to understand the function $\sin(x)$ and its properties. 2. The sine function, $\sin(x)$, is a periodic function with period $2\pi$, meaning $\sin(x + 2\pi) = \si

1. **State the problem:** Simplify the expression $$\frac{1 + \sin u}{\cos u} + \frac{\cos u}{1 + \sin u}$$. 2. **Find a common denominator:** The common denominator is $$\cos u (1

1. **State the problem:** We have a triangle with angles 70° and 61°, and the side opposite the 61° angle is 15 units. We need to find the side length opposite the 70° angle using

1. **State the problem:** We are given a triangle with angles 118°, 28°, and the remaining angle, and a side length of 5 opposite the 28° angle. We need to find the side length opp

1. **State the problem:** We are given a triangle with angles 25° and 96°, and the side opposite the 25° angle is 13 units. We need to find the length of the side opposite the 96°

1. Problem 4.1: Prove without using a calculator that $\sin 44^\circ + \sin 16^\circ = \sin 76^\circ$. Use the sum-to-product identity $\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\fra

1. **State the problem:** Prove the trigonometric identity $$\tan \theta + \cot \theta \equiv \frac{2}{\sin 2\theta}$$. 2. **Recall definitions:**

1. Let's start by stating the problem: We want to understand and derive the fundamental trigonometric equations related to triangles. 2. Consider a right triangle with an angle $\t

1. Convert from degrees to radians. (a) To convert 300° to radians, use the formula $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$.

1. Problem 9: Calculate the wingspan (length of segment DC) of the Wawa Goose given points A and B 4m apart, angles \(\angle BAC = 29^\circ\), \(\angle DAB = 81^\circ\), and \(\ang

1. Prove (a) $\cos x \tan^3 x = \sin x \tan^2 x$. Step 1: Recall that $\tan x = \frac{\sin x}{\cos x}$.