📏 trigonometry

1. **Problem statement:** Find $x$ in degrees for $0^\circ < x < 360^\circ$ given the following conditions: (i) $\sin x = -0.3782$ and $\cos x > 0$

1. **Problem statement:** Find $x$ for $0^\circ < x < 360^\circ$ given: (i) $\sin x = -0.3782$ and $\cos x > 0$.

1. **State the problem:** Simplify the expression $$\frac{x}{\cos\left(\frac{3.14}{2} - x\right)}$$. 2. **Recall the trigonometric identity:** The cosine of a difference involving

1. **State the problem:** We are given two equations:

1. نبدأ بحل السؤال الأول: إذا كان الضلع النهائي لزاوية قياسها $\theta$ يقطع دائرة الوحدة في النقطة $\left(\frac{3}{5}, -\frac{6}{5}\right)$، نعلم أن إحداثيات نقطة على دائرة الوحدة

1. **Problem Statement:** Solve the given right-angled triangles by finding all unknown sides and angles. ---

1. **Problem Statement:** Prove the given trigonometric identities using basic identities such as $\sin^2 \theta + \cos^2 \theta = 1$, $\sec^2 \theta = 1 + \tan^2 \theta$, $\csc^2

1. **Problem Statement:** Prove the given trigonometric identities using basic identities such as $\sin^2 \theta + \cos^2 \theta = 1$, $\sec^2 \theta = 1 + \tan^2 \theta$, $\csc^2

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