📏 trigonometry

1. **Problem statement:** A plane travels 200 miles at a heading of 300° from an airport. We need to find how far west of the airport the plane is. 2. **Understanding headings:** H

1. **State the problem:** We have two triangles, EBR and BRO, sharing side BR = 24.58 km. We want to find the unknown side EB in triangle EBR using given angles and sides. 2. **Ide

1. **State the problem:** A canoeist needs to travel from the lake entry point to a campsite located 2.4 km north and 3.2 km west. We want to find the direction to head directly to

1. Problem: Given $\cos \theta = -\frac{15}{17}$ and $\theta$ is in Quadrant II, find $\sin(2\theta)$. 2. Formula: Use the double-angle identity for sine:

1. The problem asks to evaluate the cosecant function at the angle $\frac{\pi}{2}$.\n\n2. Recall that $\csc \theta = \frac{1}{\sin \theta}$. This means to find $\csc \frac{\pi}{2}$

1. **State the problem:** Express $\sqrt{15} \sin(2x) + \sqrt{5} \cos(2x)$ in the form $R \sin(2x + \alpha)$.

1) Calculer, sans calculatrice : 1.a. Calculer $\cos \frac{11\pi}{12}$

1. **Problem statement:** Express $3 \cos x + 3 \sin x$ in the form $R \cos(x - a)$ where $R > 0$ and $0 < a < \frac{\pi}{2}$.

1. **Problem 6:** Find the distance between towns B and C in triangle ABC where AB = 45 km, \(\angle A = 37^\circ\), and \(\angle C = 110^\circ\). 2. Use the Law of Sines formula:

1. **State the problem:** Solve the equation $$\cos(x + \frac{\pi}{6}) + \cos\left(\frac{\pi}{6} - x\right) - \frac{3}{2} = 0.$$\n\n2. **Use the cosine sum formula:** Recall the id

1. The problem is to calculate $dA$ given by the formula $$dA = dB \frac{\sin 55^\circ}{\sin 60^\circ}$$ where $dB = 126.5$. 2. We use the sine values: $\sin 55^\circ \approx 0.819

1. **State the problem:** Two fire-lookout stations A and B are 140 miles apart, with A directly south of B. The fire is spotted with bearings N55°E from A and S60°E from B. We nee

1. **Problem Statement:** We are given a complex grid of right triangles with various angles and side lengths, some labeled with variables $x$. The goal is to find the value of $x$

1. **State the problem:** Aleena sails from port P on a bearing of 070° for 1.5 hours at 12 km/h to port Q. We need to find the bearing of port Q from lighthouse L. 2. **Calculate

1. **State the problem:** We need to find the angle $\theta$ in a right triangle where the side opposite $\theta$ is 4 units and the side adjacent to $\theta$ is 7 units. 2. **Form

1. **State the problem:** Solve the trigonometric equation $$7\cos(x) - 5\sin(x) = 5$$ for $x$. 2. **Use the formula:** We can express $a\cos(x) + b\sin(x)$ as $R\cos(x - \alpha)$

1. **State the problem:** Simplify or analyze the expression $4\sin x + 2\cos x$. 2. **Formula and rules:** We can express a linear combination of sine and cosine functions as a si

1. **Problem Statement:** Evaluate the following inverse sine expressions and express the answers in radians within the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$. 2.

1. **State the problem:** We have a right triangle with an angle of 63° and the side opposite this angle is 16. We need to find the hypotenuse $x$. 2. **Formula used:** In a right

1. The problem states that an airplane is 5,000 ft above the ground and must land on a runway 7,000 ft away horizontally. We need to find the correct trigonometric equation to calc

1. **State the problem:** We need to find the angle $\theta$ in a right-angled triangle where the side opposite $\theta$ is 3 cm and the side adjacent to $\theta$ is 8 cm. 2. **For