📏 trigonometry

1. **State the problem:** Express $\sqrt{2} \cos x - \sqrt{5} \sin x$ in the form $R \cos(x + \alpha)$ where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Then solve $\sqrt{2} \cos 2\

1. **State the problem:** We have a right triangle with hypotenuse 9.5 cm and adjacent side to angle $x$ equal to 8.3 cm. We need to find the angle $x$ in degrees, correct to 1 dec

1. The problem is to find the value of $\tan 90^\circ$. 2. Recall that the tangent function is defined as $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

1. **Problem statement:** We have two points 25 km apart east-west. A ship sails 20 km from the first point on a bearing of 045°. We need to find the bearing of the second point fr

1. **State the problem:** Town A is 30 km north of town B. A ship sails 25 km from town A on a bearing of 120°. We need to find how far east and south the ship is from town B.

1. **State the problem:** Town A is 30 km north of town B. A ship sails 25 km from town A on a bearing of 120°. We need to find how far east and south the ship is from town B.

1. **State the problem:** Town A is 30 km north of town B. A ship sails 25 km from town A on a bearing of 120°. We need to find how far east and south the ship is from town B.

1. **Problem:** Given the equation $$\frac{\sin^2 x}{\cos x} - \frac{\sin^2 x}{1+\cos x} = 1$$ for $$0^\circ \leq x \leq 360^\circ$$, find the value(s) of $$x$$. 2. **Formula and r

1. **Problem 53:** Given $\tan x = \frac{3}{4}$ and $0 < x < 90^\circ$, evaluate $\frac{\cos x}{2 \sin x}$. 2. **Recall the definitions and relationships:**

1. مسئله اول: مقدار عبارت $$A=\frac{\sin\left(\frac{\pi}{v}\right)+\sin\left(\frac{2\pi}{v}\right)+\sin\left(\frac{3\pi}{v}\right)}{\sin\left(\frac{4\pi}{v}\right)+\sin\left(\frac{

1. مسئله اول: اگر $6 = \tan x + \cot x$ باشد، مقدار عبارت $$A = \frac{\tan^7 x + \cot^7 x}{\sin^9 x + \cos^9 x}$$

1. The problem asks for the value of the angle $\theta$ when both $x$ and $y$ coordinates are negative. 2. In the Cartesian coordinate system, the angle $\theta$ is typically measu

1. **مسئله اول:** مقدار عبارت $$A = \frac{\sin \frac{7 \pi}{3} - \cos \frac{\Delta \pi}{6}}{\sin \left( \frac{-13 \pi}{3} \right) + \frac{1}{2} \tan \left( -\frac{4 \pi}{3} \right)

1. The problem is to create a table of values for the sine (sin), cosine (cos), and tangent (tan) functions. 2. These are trigonometric functions defined for an angle $\theta$ (in

1. The problem is to find the values of the trigonometric functions sine (sin), cosine (cos), and tangent (tan) for common angles. 2. The main angles we consider are $0^\circ$, $30

1. The problem asks if $\tan^{-1}(1)$ equals $\frac{\pi}{4}$.\n\n2. The function $\tan^{-1}(x)$, also called arctangent, gives the angle whose tangent is $x$.\n\n3. We know from tr

1. **Problem Statement:** We are given triangle ABC with angle $B = 120^\circ$, side $a = 12$ opposite vertex $C$, and side $c = 12$ opposite vertex $A$. We need to find the length

1. **Problem Statement:** We are given a triangle with vertices A, B, and C. The angle at vertex B is $120^\circ$. The sides opposite vertices A and C are both 12 units long, and w

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:**

1. **Prove the identity:** Given expression:

1. The problem asks for the sine of the value $$-\frac{1}{2}\sqrt{7}$$. 2. First, recognize that $$\sqrt{7}$$ is approximately 2.64575, so $$-\frac{1}{2}\sqrt{7} \approx -\frac{1}{