📏 trigonometry

1. **Problem Statement:** Sketch one full period of the function $y = 2 \sin(x) + 3$.

1. **Problem:** Find the coordinates of the point on the unit circle corresponding to an angle of $\frac{\pi}{3}$ radians. 2. **Formula and rules:** The unit circle is a circle wit

1. We start with the given formula: $$\cos(2t) = 1 - 2\sin^2(t)$$ 2. This is a double-angle identity for cosine, which relates the cosine of twice an angle to the sine squared of t

1. **Stel het probleem vast:** Los de vergelijking op $$\cos(2x) - 2 \sin^2(x) = 0$$. 2. **Gebruik trigonometrische identiteiten:** We weten dat $$\cos(2x) = \cos^2(x) - \sin^2(x)$

1. **State the problem:** Simplify the expression $5 \left( \csc \sqrt{x} \right)^3$. 2. **Recall the definition:** The cosecant function is $\csc \theta = \frac{1}{\sin \theta}$.

1. **Problem statement:** Dylan is 12 m away from a rock, and the angle of elevation to the top of the rock is 57°. We need to find the height of the rock. 2. **Formula used:** To

1. **State the problem:** We have a right triangle with sides 21, 20, and 29, where 29 is the hypotenuse. We need to find the ratio equivalent to $\sin(B)$. 2. **Recall the definit

1. **Problem statement:** Given right triangles with two sides, find $\sin$, $\cos$, and $\tan$ of angles $A$ and $B$ as common fractions. 2. **Recall:** In a right triangle, $c$ i

1. **State the problem:** We need to find the value of $\cos L$ in a right triangle $\triangle LMN$ where the right angle is at vertex $M$. The side opposite angle $L$ is $MN = 4$,

1. **Problem statement:** We need to express $\tan C$ as a fraction in simplest terms given a right triangle $CDE$ with right angle at $D$. The side opposite angle $C$ is $DE=12$,

1. **State the problem:** We need to find the length of side $x$ in a right triangle where the angle is $68^\circ$, the opposite side is $12.6$, and the adjacent side is $x$. The t

1. **Problem statement:** Given that $x$ is an acute angle and $\sin x = \frac{12}{13}$, find $\cos 2x$. 2. **Formula used:** The double-angle formula for cosine is

1. The problem is to write a trigonometric function that matches the given graph. 2. The general form of a sine function is $$f(x) = A \sin(B(x - C)) + D$$ where:

1. **State the problem:** We are given that $\sin \theta = \frac{\sqrt{2}}{2}$ and need to find $\tan \theta$. 2. **Recall the definitions and formulas:**

1. **State the problem:** We are given a right triangle with angle $x$ at vertex $B$, the side opposite $x$ is 7.4 units, and the side adjacent to $x$ is 5.6 units. We need to find

1. **State the problem:** We need to find a function of the form $f(x) = A \sec(Bx)$ that matches the given graph.

1. **State the problem:** We need to find a function of the form $f(x) = \csc(Bx - C)$ that matches the given graph. 2. **Identify key features from the graph:**

1. **State the problem:** Find the length of side $x$ in a right triangle where the hypotenuse is 13 cm and the angle adjacent to $x$ is 54° using the cosine ratio.

1. **State the problem:** A surveyor needs to find the distance between points A and B, but a house obstructs the direct path. Given are side lengths $a=58$ feet, $b=75$ feet, and

1. The problem asks to find the exact value of the sine ratio for the angle of 1.5 radians in standard position. 2. Recall that the sine of an angle in standard position on the uni

1. The problem asks to sketch an angle in standard position whose terminal arm passes through the point (1,5). 2. An angle in standard position has its vertex at the origin (0,0) a