📏 trigonometry

1. **Problem statement:** A ship sails 200 km on a bearing of 243.7 degrees. We need to find how far south and how far west the ship has traveled. 2. **Understanding bearings:** A

1. **Problem statement:** A ship sails 200 km on a bearing of 243.7 degrees. We need to find how far south and how far west the ship has traveled. 2. **Understanding the bearing:**

1. **Problem statement:** An airplane flies on a course of S 300 E for 150 km. We need to find how far south the plane is from its starting point. 2. **Understanding the direction:

1. The problem is to simplify or understand the expression $\sin^2 x$. 2. The notation $\sin^2 x$ means $(\sin x)^2$, which is the square of the sine of $x$.

1. **State the problem:** We need to prove the trigonometric identity $$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$. 2. **Recall the sine difference fo

1. **Stating the problem:** Find the value of $\sin(\tan^{-1}(x))$ or understand the relationship between sine and tangent functions. 2. **Formula and rules:** Recall that $\tan(\t

1. **Stating the problem:** Prove or verify the identity:

1. The problem is to find the angle in degrees. 2. To find an angle in degrees, you typically use trigonometric functions or convert from radians.

1. مسئله: بررسی تساوی الف) \( A = \sin^{-1} x + \sin^{-1} y = \sin^{-1} \left( x \sqrt{1-y^2} + y \sqrt{1-x^2} \right) \) با شرط \( |\sin^{-1} x + \sin^{-1} y| \leq \frac{\pi}{2} \

1. **State the problem:** Prove the identity $$\frac{\cos A}{\sin A + \cos B} + \frac{\cos B}{\sin B - \cos A} = \frac{\cos A}{\sin A - \cos B} + \frac{\cos B}{\sin B + \cos A}$$

1. **State the problem:** Prove the identity $$\frac{\cos A}{\sin A + \cos B} + \frac{\cos B}{\sin B - \cos A} = \frac{\cos A}{\sin A - \cos B} + \frac{\cos B}{\sin B + \cos A}$$

1. **State the problem:** Solve the trigonometric equation $$6 \tan x \sin x = 5 \sin x - \cos x$$ for $$0^\circ \leq x \leq 360^\circ$$. 2. **Rewrite the equation:** Recall that $

1. **Problem:** Solve the trigonometric equation $$9 \sin^2 x + 3 \cos x - 7 = 0$$ for $$0^\circ \leq x \leq 360^\circ$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^

1. **Problem statement:** A peacock is sitting on top of a tree 10 m high and observes a snake moving on the ground. The snake is $10\sqrt{3}$ m away from the base of the tree. We

1. Planteamos el problema: Demostrar la identidad $$\frac{\cos \phi + \cos \phi \sen \phi}{1 + \sen \phi} = \cos \phi$$. 2. Observamos que el numerador tiene un factor común $$\cos

1. Given that $\sin \theta = \frac{5}{13}$ and $\theta$ is acute, find: (a) $\tan \theta$

1. **Problem statement:** An airplane leaves Manila and flies 300 km on a bearing N 55° W, then flies 200 km on a bearing S 25° W. We need to find the distance from Manila after th

1. **Problem Statement:** Find the sum of the series $\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \cdots + \cos 180^\circ$.\n\n2. **Formula Used:** The sum of cosines of an arithm

1. **State the problem:** Given that $\sec 2A = \csc (A - 27^\circ)$ and $2A$ is an acute angle, find the measure of $\angle A$. 2. **Recall the definitions:**

1. **State the problem:** Given that $\tan \theta + \cot \theta = 5$, find the value of $\tan^2 \theta + \cot^2 \theta$. 2. **Recall the formula:** We know that

1. **State the problem:** Given that $\tan \theta + \cot \theta = 5$, find the value of $\tan 2\theta + \cot \theta$. 2. **Recall formulas and identities:**