📏 trigonometry

1. **State the problem:** We are given the function $$y = -3 \cos x$$ and want to understand its properties. 2. **Recall the cosine function:** The basic cosine function $$\cos x$$

1. **Problem Statement:** We want to understand the transformation of the graph of the sine function given by the equation $$y=\frac{1}{2}\sin x$$. 2. **Formula and Rules:** The ge

1. **State the problem:** We have points $A$, $B$, and $C$ on level ground with given bearings and distances:

1. **Problem Statement:** We have a right triangle with one leg measuring $15$ ft and the hypotenuse measuring $25$ ft. We need to find the length of the other leg.

1. The problem is to simplify the expression $\frac{\sin(x)}{\cos(x)}$. 2. Recall the trigonometric identity: $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

1. **State the problem:** We need to find the exact value of $\csc M$ in simplest form for the right triangle $\triangle MKL$ with right angle at $L$. 2. **Identify sides relative

1. **State the problem:** We need to find the exact value of $\csc M$ in the right triangle $KML$ where $\angle L$ is the right angle. 2. **Identify the sides relative to angle $M$

1. **State the problem:** Verify the trigonometric identity: $$(\sin x - \cos x)^2 = 1 - \sin 2x$$

1. **State the problem:** Find the exact value of the expression $$2 \cos^2(-150^\circ) - 1$$. 2. **Recall the formula:** The expression resembles the double-angle identity for cos

1. The problem is to simplify or analyze the expression $\sin x + \cos x$. 2. A useful formula to rewrite sums of sine and cosine is the amplitude-phase form:

1. المشكلة: حساب سكس رؤسي (الجيب التمام للرأس) في مثلث قائم الزاوية. 2. القاعدة: في مثلث قائم الزاوية، سكس رؤسي (cosine) هو نسبة طول الضلع المجاور للرأس إلى طول الوتر.

1. **Problem statement:** We have points R, S, and T on horizontal ground with RS = 9.8 m and ST = 13.5 m. A vertical flagpole RQ stands at R. The angle of depression from Q to S i

1. **State the problem:** Given an acute triangle $\triangle ABC$ with $\angle B = 58.8^\circ$, side $c = 10.3$ cm (opposite $\angle C$), and side $b = 10.5$ cm (opposite $\angle B

1. **State the problem:** We have two curves:

1. **State the problem:** Solve the equation $$\sin^2\theta - \cos^2\theta - \cos\theta + 1 = 0$$ for $$0 \leq \theta < 2\pi$$. 2. **Use the Pythagorean identity:** Recall that $$\

1. **Problem:** In triangle ABC, given angles $A=42^\circ$, $B=58^\circ$, and side $a=15$ cm, find sides $b$ and $c$. 2. **Formula:** Use the Law of Sines: $$\frac{a}{\sin A} = \fr

1. **State the problem:** Given triangle ABC with angles $A=42^\circ$, $B=58^\circ$, and side $a=15$ cm opposite angle $A$, find sides $b$ and $c$.

1. Problem: Find the trigonometric ratios for given angles using the unit circle. 2. Formula: On the unit circle, coordinates at angle $\theta$ are $(\cos \theta, \sin \theta)$.

1. Let's start by understanding the problem: we want to solve a trigonometry problem using cosine. 2. The cosine of an angle in a right triangle is the length of the adjacent side

1. **Stating the problem:** Solve the given trigonometry problem involving a triangle. 2. **Formula and rules:** Use the basic trigonometric ratios: sine, cosine, and tangent.

1. **State the problem:** Find the value of $\sin(\arccos(\frac{15}{17}))$. 2. **Recall the relationship:** If $\theta = \arccos(x)$, then $\cos(\theta) = x$ and $\sin(\theta) = \s