📏 trigonometry

1. لنبدأ بتعريف الدوال المثلثية: هي دوال تربط بين زوايا المثلثات وأطوال أضلاعها. 2. الدوال الأساسية هي: الجيب $\sin(\theta)$، جيب التمام $\cos(\theta)$، والظل $\tan(\theta)$.

1. **Prove** $\frac{1}{\sec \delta + \tan \delta} = \frac{1 - \sin \delta}{\cos \delta}$. Start with the left side:

1. **Problem:** Prove the identity \(\frac{\tan^2 C - 1}{1 + \tan^2 C} = 1 - 2 \cos^2 C\). 2. Start with the left side (LHS):

1. **Problem statement:** Find the exact values of the following trigonometric functions: (a) $\tan\left(\frac{\pi}{3}\right)$

1. **State the problem:** We need to find two angles related to a cinema seating arrangement:

1. **State the problem:** We have a cinema seating layout with rows A and P and a screen. We want to find:

1. **State the problem:** A robin flies 21 km at a bearing of 039° from its starting point, then flies 36 km due south. We need to find how far south of its starting point the robi

1. **Problem c)** Solve for $\theta$ given $\tan \theta = \frac{1}{\sqrt{3}}$ for $-360 \leq \theta \leq 360$. - Recall that $\tan \theta = \frac{1}{\sqrt{3}}$ corresponds to angle

1. The problem is to solve the trigonometric equation $$\sin(x) = \frac{1}{2}$$ for all solutions. 2. From the unit circle, we know that $$\sin(x) = \frac{1}{2}$$ at two principal

1. **Problem statement:** In triangle $\triangle PQR$, given $\angle Q = 42^\circ$, side $PR = 12$ cm, and side $PQ = 10.2$ cm, find: (i) $\angle R$

1. The problem is to convert the angle $\frac{2 \pi}{3}$ radians into degrees. 2. Recall the conversion formula between radians and degrees:

1. The problem is to convert an angle of 150° to radians. 2. Recall the conversion formula between degrees and radians: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

1. You asked to work with angles in radians. 2. Radians are a way to measure angles based on the radius of a circle.

1. The problem is to find the value of $\cos\left(\frac{\sqrt{11}}{6}\right)$.\n\n2. Here, $\frac{\sqrt{11}}{6}$ is the angle in radians. We need to evaluate the cosine of this ang

1. **State the problem:** Given $\cos \theta = \sqrt{\frac{11}{6}}$ and $\theta$ is in Quadrant I, find the exact value of $\sin \theta$ in simplest form. 2. **Recall the Pythagore

1. We are given that $\cos \theta = -\frac{1}{6}$ and that $\theta$ is in Quadrant III. 2. In Quadrant III, both sine and cosine are negative.

1. **State the problem:** Given $\cos \theta = -\frac{2}{3}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin \frac{\theta}{2}$, $\cos \frac{\theta}{2}$, and $\tan \frac{\theta}{2}$

1. **Evaluate the expression:** $$8 \sin 45^\circ - \sin 60^\circ \cdot \cos 30^\circ - \frac{1}{4} \tan 45^\circ$$ 2. **Recall exact trigonometric values:**

1. **Problem 3.4:** Determine the angle $\theta$ formed by the line from the origin to the point $(3, -4)$ with the positive x-axis, correct to 2 decimal places. 2. The point $(3,

1. The problem involves understanding the inverse tangent function, often written as $\tan^{-1}(x)$ or $\arctan(x)$, which gives the angle whose tangent is $x$. 2. To find $\arctan

1. **Problem:** Find $\sin^{-1}(\frac{1}{2})$. Step 1: Recall that $\sin^{-1}(x)$ is the angle whose sine is $x$.