📏 trigonometry

1. Let's state the problem: We want to understand and apply the cosine rule and sine rule to solve triangles. 2. The cosine rule states: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $a$,

1. **Problem 1: Find height of the tree.** Given: $d=20$ m, angle of elevation $\theta=30^\circ$.

1. **Problem Statement:** We have three right-angled triangles with given sides and angles, and we need to find the unknown side $x$ in each case. 2. **Formulas and Rules:** In rig

1. **Problem statement:** We have three right triangles and need to find the unknown side $x$ in each case using trigonometric ratios. 2. **Recall the trigonometric definitions:**

1. **Problem Statement:** Calculate the unknown side length $x$ in each right-angled triangle given the angles and sides, correct to 4 significant figures. 2. **Key formulas and ru

1. Problem 4: Given $\tan X = \frac{15}{112}$, find $\sin X$ and $\cos X$ in fraction form. 2. Recall the identity for tangent: $\tan X = \frac{\sin X}{\cos X}$.

1. **Problem 1:** Given a right triangle XYZ with \(\angle Y = 90^\circ\), \(XY = 9\) cm, and \(YZ = 40\) cm, find \(\sin Z\), \(\cos Z\), \(\tan X\), and \(\cos X\). 2. **Step 1:*

1. **Problem Statement:** A lighthouse 50 m tall observes a ship moving from point P to Q. The angles of depression from the top of the lighthouse to the ship are 30° at P and 45°

1. **Problem statement:** We need to find the height of a tree given two angles of elevation (60° and 30°) observed from two points along a horizontal line, where the second point

1. **Problem statement:** We have a right-angled triangle with an angle $\theta$ such that $\tan \theta = 5$. The side opposite to $\theta$ is labeled $x$, and the vertical side ad

1. **Problem statement:** Ibraheem hikes from Gator Swamp to Champion Lookout. Gator Swamp is 4 km from Old Town Road along Route 67 at a bearing of 15°. The hiking trail from Gato

1. **State the problem:** Given the equation $m^2 - n^2 = 4\sqrt{mn}$ and $m = \tan A + \sin A$, prove that $n = \tan A - \sin A$.

1. **Stating the problem:** We are given the equation $m^2 - n^2 = 4\sqrt{mn}$ and the expression $m = \tan A + \sin A$. We need to prove that $n = \tan A - \sin A$. 2. **Recall th

1. **Problem 1: Distance the second ship must sail** A ship travels west 50 km, then sails 52.4 km in direction E25°N. We want the direct distance from the harbour to the ship's fi

1. **Problem statement:** A tree is broken by the wind such that the broken part makes a 45° angle with the ground. The distance from the root to the point where the top touches th

1. **State the problem:** We have triangle ADE with sides AD = 60, AE = 146, and angle DAE = 55°. We need to find: (a) The bearing of A from E.

1. Δίνεται ότι $\tan \theta = \frac{4}{3}$ και $180^\circ < \theta < 270^\circ$. Θέλουμε να υπολογίσουμε την τιμή της παράστασης: $$A = \frac{4\cos \phi \theta - 5\sin \theta}{3\ta

1. **State the problem:** Find the smallest positive root of the equation $$\sin 3x = -1$$ where $x$ is in degrees. 2. **Recall the sine function properties:** The sine function eq

1. **Problem Statement:** You are on a sailing trip and want to find the angle of elevation to the top of a lighthouse from your boat. The triangle formed has two sides measuring 1

1. **State the problem:** We need to graph the function $f(x) = \csc x$ and discuss its behavior near $x=0$. 2. **Recall the definition:** The cosecant function is defined as $\csc

1. **Problem statement:** Sketch the function $f(x) = \cot(x)$ and mark its points of discontinuity. 2. **Formula and definition:** The cotangent function is defined as $\cot(x) =