📏 trigonometry

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:**

1. **Problem statement:** Calculate the value of $\theta$ in the first triangle where the opposite side to $\theta$ is 13 and the hypotenuse is 19. 2. **Formula:** Use the sine rat

1. **Problem:** Calculate the value of $\theta$ to the nearest hundredth using trigonometric ratios and inverse trig functions for the first triangle with angle 45°, leg $x$, and h

1. **Problem statement:** The shadow of a skyscraper is 100 m longer when the angle of elevation of the sun is 40° than when it is 60°. We need to find the height of the skyscraper

1. The problem is to understand how to use the number 12.5 to get 34 degrees. 2. This likely involves a trigonometric function where 12.5 is related to an angle of 34 degrees.

1. **State the problem:** Calculate the measure of angle $\theta$ to the nearest degree using the given triangle and trigonometric relationships. 2. **Recall the tangent function:*

1. **Problem:** Prove the identity $\cot x (\cot x + \tan x) = \csc^2 x$. 2. **Recall the definitions and identities:**

1. **Problem statement:** A tree broken by the wind forms a right-angled triangle with the ground. The broken part of the tree makes an angle of 60° with the ground. The top of the

1. **Stating the problem:** Solve the equation $$6 \cot x \left(1 + \cot^2 x\right) = \frac{3\sqrt{3}}{2} \sin 2x$$ for $$0 \leq x \leq 2\pi$$. 2. **Recall identities:**

1. The problem is to solve the equation $$\sin\left(A + \frac{\pi}{2}\right) = \cos\left(\frac{A}{2}\right) + 2$$ for $$0 \leq A \leq 2\pi$$. 2. Recall the identity $$\sin\left(x +