📏 trigonometry

1. **State the problem:** Given $m\angle A = 85^\circ$, side lengths $a = 15$, $b = 25$, and $h = b \sin A$, classify the triangle as acute or obtuse, find the number of solutions,

1. **Problem Statement:** Given a triangle with two sides and a non-included angle (SSA), use the Law of Sines to find the unknown sides or angles. 2. **Law of Sines Formula:**

1. **State the problem:** Solve and verify the trigonometric identity:

1. The problem is to solve the equation $\sin x = 0.45$ for $x$. 2. Recall that the sine function gives the ratio of the opposite side to the hypotenuse in a right triangle, and it

1. The problem is to simplify or evaluate the expression $\cos(x)^2 - 0.5$. 2. Recall the Pythagorean identity: $$\cos^2(x) + \sin^2(x) = 1$$ and the double-angle formula for cosin

1. **Problem statement:** We have a right triangle with one side length $9.5$ inches and an opposite angle of $26^\circ 45'$ with a possible error of $15'$.

1. **State the problem:** Convert the angle 405° from degrees to radians. 2. **Formula:** To convert degrees to radians, use the formula:

1. The problem is to convert the angle 315° from degrees to radians. 2. The formula to convert degrees to radians is:

1. **State the problem:** Convert the angle $\pi$ radians to degrees. 2. **Formula:** To convert radians to degrees, use the formula:

1. The problem is to convert an angle from radians to degrees. 2. The formula to convert radians to degrees is:

1. The problem is to convert the angle $2\pi$ radians into degrees. 2. The formula to convert radians to degrees is:

1. **State the problem:** Convert the angle $\frac{7\pi}{4}$ radians to degrees. 2. **Formula:** To convert radians to degrees, use the formula:

1. The problem asks to convert an angle of $-4.5$ radians to degrees, rounding to the nearest tenth. 2. The formula to convert radians to degrees is:

1. **Problem statement:** We have two right-angled triangles:

1. **Stating the problem:** We are given a triangle with sides 7 cm, 9 cm, and 12 cm, and we need to find the size of the angle opposite the 7 cm side. 2. **Formula used:** To find

1. **Problem:** A scuba diver is 145 feet above a sunken ship and measures the angle of depression to the ship as 73°. Find the distance the diver needs to swim to reach the ship,

1. **Problem statement:** Find the measure of the indicated angle in each right triangle using the given sides. 2. **Formula and rules:** In a right triangle, use trigonometric rat

1. The problem involves understanding the laws of depression and elevation, which relate to angles formed when looking up or down from a horizontal line. 2. The law of elevation st

1. **Problem:** A ship travels from A to B (3 km on a bearing of 040°) and then from B to C (5 km on a bearing of 120°). Find the distance AC. 2. **Formula and rules:** To find the

1. The problem asks to find the value of $\cos \theta$ in a right-angled triangle $MNP$ where $MN = \sqrt{7}$ cm and $MP = 4$ cm, with the right angle at $N$ and angle $\theta$ at

1. **State the problem:** We need to sketch the graph of $y = |\cos(\frac{x}{4})|$ for $0 \leq x \leq 8\pi$ and solve the equation $|\cos(\frac{x}{4})| = \frac{1}{2}$ in the same i