📏 trigonometry

1. **State the problem:** We want to express the product $\cos x \sin 7x$ as a sum or difference of trigonometric functions. 2. **Recall the product-to-sum formula:**

1. **State the problem:** Adrian flies $357$ km due south from Darwin, then flies on a bearing of $N29^\circ W$ until he is due west of Darwin. We need to find how far Adrian is fr

1. **Problem Statement:** We have two circles touching a straight line at points $X$ and $Y$ respectively, and a common tangent line passing through their top points. Given $OX = 6

1. **Problem:** Find the value of $\cos 105^\circ$ without using a calculator and express it in the form $\frac{1-\sqrt{a}}{b\sqrt{b}}$. 2. **Formula:** Use the cosine addition for

1. **Problem statement:** We need to find the length of the hypotenuse $m$ in a right triangle where the vertical side adjacent to the $50^\circ$ angle is $17$ units. 2. **Formula

1. **State the problem:** We need to find the length $m$ of the vertical side in a right triangle where the hypotenuse is $26$ and the angle opposite to $m$ is $43^\circ$. 2. **Rec

1. **Problem Statement:** We need to find the angle $\theta$ formed between the vertical pole and the upper arm of the streetlight. 2. **Given:**

1. **State the problem:** We need to evaluate the expression $$P = \frac{1}{\pi} \cos^{-1}\left(\frac{25.8}{\sqrt{26.8 \times 25.2}}\right)$$. 2. **Recall the formula and rules:**

1. **Problem statement:** We have a right triangle with a right angle at the bottom-left, a 30° angle at the top, the bottom horizontal side length is $4$, the vertical leg is $B$,

1. **State the problem:** Convert $5$ radians to degrees and round to the nearest whole number. 2. **Formula:** To convert radians to degrees, use the formula:

1. **State the problem:** We are given a triangle with angles $A = 34^\circ$, $B = 27^\circ$, and side $b = 11$. We need to find side $c$ opposite angle $C$, and angle $C$ itself.

1. The problem is to find the inverse of the function \( \cos(\theta) = \frac{1}{3} \).\n\n2. The inverse cosine function, denoted as \( \cos^{-1}(x) \) or \( \arccos(x) \), gives

1. **State the problem:** Solve the equation $$3\sin^2 2peter + 8\cos 2peter = 0$$ for $$peter$$ in the interval $$0^\circ \leq peter \leq 180^\circ$$. 2. **Use the Pythagorean ide

1. **State the problem:** We need to express $\tan 75^\circ$ in the form $a + b\sqrt{3}$ where $a$ and $b$ are integers. 2. **Recall the formula for tangent of sum of angles:**

1. **State the problem:**\nWe are given the depth of water $D$ at Southview harbour on 18th October 2017 modeled by\n$$D = 6 + 2 \sin \left(\frac{4\pi t}{25}\right) - 1.5 \cos \lef

1. **State the problem:** We have a vertical cliff with two boats on opposite sides. The angles of depression to the boats are $30^\circ$ and $60^\circ$, and the distance between t

1. **State the problem:** We have a vertical tower with a flagstaff of length $10$ m on top. From a point on level ground, the angle of elevation to the top of the tower is $45^\ci

1. **Stating the problem:** (a) Express $\sec^2 A - 1$ in terms of $\cos 2A$.

1. **State the problem:** We are given a triangle $\triangle ABC$ with the ratio of sines of its angles as $\sin A : \sin B : \sin C = 4 : 5 : 6$. We need to find $\cos(A + C)$. 2.

1. **State the problem:** A camel travels from an oasis at a bearing of $104^\circ$. It stops $23$ km east of the oasis. We need to find how far south it stopped. 2. **Understandin

1. **Problem Statement:** Calculate the bearing of point Y from point Z given the triangle with angles 135° at X and 105° at Y. 2. **Understanding Bearings:** Bearings are measured