📏 trigonometry

1. **State the problem:** We are given $$\sin \theta = -\frac{3}{4}$$ and $$\cos \theta < 0$$. We need to find $$\cos \theta$$, $$\tan \theta$$, $$\csc \theta$$, $$\sec \theta$$, a

1. **Problem Statement:** Two helicopters start from the same point. The first flies 5 km on a bearing of $60^\circ$, the second flies 7 km on a bearing of $130^\circ$. We need to

1. **State the problem:** We need to analyze the function $$y = 4\cos x + 3\sin x$$ for $x$ from $0^\circ$ to $270^\circ$ in $30^\circ$ intervals.

1. **Planteamiento del problema:** Calcular $\sin A$, $\cos A$ y $\tan A$ para dos triángulos rectángulos dados.

1. **Problem:** Determine which of the following trigonometric identities is not true. Given options:

1. **Problem:** A fisherman has a triangular pond with the longest side measuring $35$ feet and two angles measuring $40^\circ$ and $55^\circ$. Find the lengths of the other two si

1. Verificar la identidad: $$\cot^2 x \cdot \csc x = \csc x - \sin x$$ - Recordemos que $$\cot x = \frac{\cos x}{\sin x}$$ y $$\csc x = \frac{1}{\sin x}$$.

1. **State the problem:** We need to find the value of $x$ given the equation $25\tan60^\circ = x$. 2. **Recall the formula:** The tangent of an angle in a right triangle is the ra

1. **Problem:** Use the sine rule to find $x$ in the triangle with sides and angles given: side opposite $49^\circ$ is $x$, side opposite $62^\circ$ is $13.9$ cm. 2. **Formula:** T

1. **Problem 1: Area of the kite** We are given a kite-shaped quadrilateral with diagonals measuring $25$ cm and $40$ cm, and a vertical height of $60$ cm (likely the length of one

1. **State the problem:** Solve the equation $$81 \sin^2 x + 81 \cos^2 x = 30$$ for $$x$$ in the interval $$0 \leq x \leq 360^\circ$$.

1. **Problem Statement:** We have a right-angled triangle PQR with right angle at Q. Given sides: $PQ = 9$, $QR = 12$, and hypotenuse $PR = 15$.

1. **Problem statement:** Calculate the vertical height $YZ$ of the signboard above the ground given the hypotenuse $XY=12$ m and angle $\angle X=65^\circ$.

1. **State the problem:** We are given that $\tan x + \tan y + \tan z = \pi$ and need to show that $x + y + z = xyz$.

1. Problema 01: Calcular $$ (\sec^2 45^\circ + \tan 45^\circ) \cdot \cot 37^\circ - 2 \cos 60^\circ $$. 2. Recordemos valores trigonométricos importantes:

1. **Problema 01:** Calcular $$ (\sec^2 45^\circ + \tan 45^\circ) \cdot \cot 37^\circ - 2 \cos 60^\circ $$ 2. **Fórmulas y valores importantes:**

1. **State the problem:** We are given that $$\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi$$ and we need to prove that $$x + y + z = xyz$$.

1. **State the problem:** We need to sketch the graph of the function $y= \cos(2x)$. 2. **Recall the formula and properties:** The general cosine function is $y= \cos(bx)$ where $b

1. **Problem 1.1:** Evaluate $\frac{\cos 15^\circ \times \tan 30^\circ \times \sin 315^\circ}{\sin 30^\circ \times \cos 210^\circ \times \sin 105^\circ}$ without a calculator. 2. *

1. **Problem Statement:** A soccer player S is 15 m from the back line CH. The goal posts FG and EH are 7.32 m apart. The angle FSG (angle at S between the left goal post F and the

1. **State the problem:** We need to prove that $$(\sin x + \cos x)^2 + (\sin x - \cos x)^2 = 2$$ for any angle $x$. 2. **Recall the formula:** The square of a sum and difference a