📏 trigonometry

1. Simplify $\frac{\tan A (\sec A - \cos A)}{\sin A}$. Recall the identities: $\tan A = \frac{\sin A}{\cos A}$ and $\sec A = \frac{1}{\cos A}$.

1. **State the problem:** We have a triangle with sides 5.4 cm and 4.4 cm, and an angle of 40° opposite the 5.4 cm side. We need to find the angle $x$ opposite the 4.4 cm side.

1. **State the problem:** We need to find the perimeter of a right-angled triangle where one angle is $32^\circ$ and the side adjacent to this angle is $16.6$ m. 2. **Identify the

1. **State the problem:** We have a right triangle with hypotenuse length 20 and a leg opposite the missing angle measuring 13. We need to find the measure of the missing angle. 2.

1. **State the problem:** We need to find the angles $m\angle B$, $m\angle A$, $m\angle W$, and $m\angle H$ given their sine, cosine, or tangent values. 2. **Formula used:** To fin

1. The problem is to understand why in step 5 of a certain derivation, the expression $bd = r \sin \theta$ holds. 2. Typically, in polar coordinates or trigonometric contexts, $r$

1. The problem is to identify the sine and cosine graphs that correspond to a given function or context. 2. The sine function is defined as $y = \sin(x)$ and the cosine function as

1. **Problem Statement:** You are on a Ferris wheel with diameter 40 feet, taking 8 seconds for one full revolution. Your height above the ground varies sinusoidally with time $t$

1. The problem asks why $\cos \theta$ is expressed as $\sqrt{1 - \sin^2 \theta}$. 2. This comes from the Pythagorean identity in trigonometry, which states:

1. Problem 5: Use the Pythagorean Theorem to find the length of side $b$ (hypotenuse $AC$) in triangle $ABC$ where $AB=6$ cm and $BC=8$ cm. 2. The Pythagorean Theorem states: $$a^2

1. The problem states that the angle $t$ corresponds to the point $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ on the unit circle. 2. Recall that on the unit circle, the

1. **Problem Statement:** Given a circle centered at the origin and a point on the circle at coordinates $(10, 24)$, find:

1. **Problem statement:** Find all values of $t \in [0, 2\pi]$ that satisfy the following equations: (a) $\sin(t) = -\frac{1}{2}$

1. **State the problem:** We need to determine the exact values of $\sin\left(\frac{2}{3}\pi\right)$ and $\cos\left(\frac{2}{3}\pi\right)$ and check if the given decimal approximat

1. The problem is to determine which of the points A, B, C, or D lie on the graph of the function $f(x) = \tan(5x)$. 2. To check if a point $(x,y)$ lies on the graph, we substitute

1. **State the problem:** Find $\sin(B)$ for a right triangle where the side opposite angle $B$ is $\sqrt{13}$, the adjacent side is $6$, and the hypotenuse is $7$. 2. **Recall the

1. **State the problem:** We have a triangle with angles 60°, 75°, and 45°, and the side opposite the 60° angle is 240. We want to find the length of side $x$, which is opposite th

1. **Problem Statement:** Solve for the distances to the ships offshore given the lighthouse height of 350 feet and angles of depression $\theta$ and $\beta$.

1. **Problem Statement:** We have a lighthouse 350 feet above sea level. Two angles of depression from the lighthouse to two ships offshore are given as $\theta$ (upper angle) and

1. **State the problem:** Solve the equation $\sin 2x = 2 \sin x$. 2. **Recall the double-angle formula:** $\sin 2x = 2 \sin x \cos x$.

1. **State the problem:** Solve the equation $$\cos 2x + \cos x = 0$$ for $$x$$ in the interval $$[0, 2\pi)$$. 2. **Use the double-angle formula:** Recall that $$\cos 2x = 2\cos^2