📏 trigonometry

1. **State the problem:** We need to find the smallest possible angle $\theta$ between the plane's path and the ground, given the plane reaches a height of 10 km and flies a distan

1. **Problem Statement:** Determine whether to use the Law of Sines or Law of Cosines to find the indicated side length or angle measure in each triangle. 2. **Formulas:**

1. **Problem:** A construction worker looks up at the top of a crane 60 meters tall. The angle of elevation to the top is 35 degrees. Find the distance from the worker to the base

1. **State the problem:** Simplify the expression $$(\tanh x - 1)(\tanh x + 1) - 2 \tanh^2 x + \sec^2 x$$ using trigonometric identities. 2. **Recall relevant identities:**

1. **State the problem:** Simplify the expression $$\cos^2 m (\tan m - 1)(\tan m + 1) - 2 \tan^2 m + \sec^2 m$$. 2. **Recall formulas and identities:**

1. **Problem Statement:** Find the sine, cosine, and tangent of the given angles using the quadrant system. 2. **Formula and Rules:**

1. **Problem:** Given the diagram of △ABC with angles B = 50°, A = 80°, and side BC = 20 units, find the value of sec A. **Step 1:** Recall that \(\sec A = \frac{1}{\cos A}\).

1. **State the problem:** We want to analyze the function $$y = \sin(3x) \cos(2x)$$ and understand its behavior. 2. **Formula and identities:** We can use the product-to-sum identi

1. The problem is to find the value of $\cot 42^\circ$. 2. Recall that $\cot \theta = \frac{1}{\tan \theta}$, so we can find $\cot 42^\circ$ by calculating $\tan 42^\circ$ and then

1. The problem is to find the value of $\tan 42^\circ$. 2. Recall that $\tan \theta = \frac{\sin \theta}{\cos \theta}$, but for specific angles like $42^\circ$, we typically use a

1. Solve the trigonometric equation: $5\sin\phi + 3 = 0$ for $0^\circ \leq \phi \leq 360^\circ$. 2. Rearrange the equation to isolate $\sin\phi$:

1. **Problem Statement:** From the top of a rock 100 m high, the depression angles to the top and base of a tower are 22° and 33° respectively. The base of the rock and the tower a

1. **State the problem:** We are given the function $y = 2 \sin\left(\frac{\pi}{4}(x + 3)\right) + 1$ and need to analyze its properties. 2. **Formula and explanation:** The genera

1. **State the problem:** We have a right-angled triangle with a hypotenuse of length 27.9 cm and an angle of 53° adjacent to the side of length $n$. We need to find the length $n$

1. **Problem statement:** Given that $A+B+C=180^\circ$, prove that $$\cos^2 A + \cos^2 B - \cos^2 C = 1 - 2 \sin A \sin B \cos C.$$\n\n2. **Recall the identity:** Since $A+B+C=180^

1. **Problem Statement:** We need to find the final position of a fishing boat that traveled in two legs with given bearings and distances, then calculate:

1. **Problem statement:** Find angle $A$ in a triangle where side $b=28$, side $a=29$, and angle $C=52^\circ$. We can use the Law of Cosines or Law of Sines. 2. **Choosing the form

1. **Problem statement:** Find angle $A$ in a triangle where side $b=28$, angle $C=52^\circ$, and side $a=29$. 2. **Formula used:** We use the Law of Sines which states:

1. The problem is to simplify the expression $\csc^2 x + \sec^2 x$. 2. Recall the Pythagorean identities:

1. **Stating the problem:** Prove that $$\tan A + 2 \tan^2 A + 4 \tan^4 A + 8 \cot^8 A = \cot A.$$\n\n2. **Recall definitions and identities:** \n- $\tan A = \frac{\sin A}{\cos A}$

1. The problem is to simplify or solve an expression involving $\cos 70^\circ$, $\cos 20^\circ$, and $\cos 25^\circ$ without directly knowing their values. 2. We use trigonometric