📏 trigonometry

1. **Problem statement:** An aircraft flies from A to B, distance AB = 600 km. Due to bad weather, it detours via C. The line CA makes an angle of 28° 15' with AB, and CA = 500 km.

1. The term "basic angle" typically refers to the smallest positive angle between the terminal side of a given angle and the x-axis in standard position. 2. It is often used in tri

1. The problem is to find the value of $\sin 10^\circ$. 2. Since $10^\circ$ is not one of the standard angles with simple sine values, we can use the sine addition formula or appro

1. Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. 2. The primary functions in trigo

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:**