📏 trigonometry

1. Stating the problem: We have two airplanes observed from the same point on the ground. The higher plane is 6000 m above the ground. The angles of elevation to the two planes are

1. Problem 129.a: Simplify $\sqrt{2} \sin\left(\frac{\pi}{4} + \alpha\right) - \sin \alpha$.\n Using the sine addition formula, $\sin(a+b) = \sin a \cos b + \cos a \sin b$, we get:

1. We are asked to find the value of $x$ given that the maximum value of $\prod_{i=1}^n \cos \alpha_i$ under the conditions $0 \leq \alpha_i \leq \frac{\pi}{2}$ and $\prod_{i=1}^n

1. We are asked to find the smallest positive integer $x$ such that $$\tan(x - 160) = \frac{\cos 50}{1 - \sin 50}.$$ 2. Notice the right side: it resembles the tangent half-angle i

1. The problem is to analyze the function $y = \cos x$ and understand its key features including intercepts and extrema. 2. The cosine function has the form $y = \cos x$, which is

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$

1. Convert from degrees to radians. (a) To convert $300^\circ$ to radians, use the formula:

1. **State the problem:** Find the smallest positive integer $x$ (in degrees) such that $$\tan(x - 160^\circ) = \frac{\cos 50^\circ}{1 - \sin 50^\circ}.$$\

1. Stating the problem: Find the exact values of the following trigonometric functions: (a) $\tan(\frac{\pi}{3})$

1. Convert degrees to radians. (a) Given $300^\circ$, use the formula $\text{radians} = \text{degrees} \times \frac{\pi}{180}$.

1. **Problem statement:** Prove the following identities:

1. Convert from degrees to radians: (a) The formula to convert degrees to radians is $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$.

1. **Prove** $\tan \theta \sin \theta + \cos \theta = \sec \theta$. Start from the left side (LHS):

1. **State the problem:** Jimmy is on a ship 708 meters away from the base of a cliff. The angle of elevation to the top of the lighthouse on the cliff is 26 degrees. We need to fi

1. Problem statement: Jimmy is on a ship 708 meters away from the base of a cliff. The angle of elevation from Jimmy to the top of the lighthouse on the cliff is $26^\circ$. 2. We

1. Stating the problem: We have a light post 5 m tall casting a shadow of 9.3 m. We need to find the angle of elevation of the sun at that time, denoted as $\theta$, rounded to two

1. **State the problem:** Eileen measures the angle of elevation to the top of a tree to be 42° from a point 27 metres away from the base. Find the height $h$ of the tree.

1. **Problem Statement:** Find the lengths of the sides labeled $y$ and $z$ in two right-angled triangles where each triangle has a given side length adjacent to a given angle.

1. **Problem (a):** Calculate the length of the side $w$, opposite the $15^\circ$ angle, in a right triangle with hypotenuse $10$ cm. 2. Use the sine function because sine relates

1. Let's clarify the problem: You have given two angle measures, 298° and 298.165°, and you mention two rectangles stacked vertically with quotation marks. The question seems uncle

1. The problem is to convert the angle 336° 44' 33'' to decimal degrees. 2. Recall that 1 degree ($1^\circ$) equals 60 minutes (') and 1 minute equals 60 seconds ('').