📏 trigonometry

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

1. **Problem statement:** Calculate the value of $$\sin^2 35^\circ + \sin^2 10^\circ + \sqrt{2} \sin 35^\circ \sin 10^\circ$$. 2. **Formula and rules:** Recall the identity for sin

1. **State the problem:** Verify the trigonometric identities and solve the given trigonometric questions. 2. **Identity verification:**

1. **Problem statement:** Find the angle $\theta$ for point A(-3,4) in the domain $0 < \theta \leq 4\pi$. 2. **Formula and rules:** The angle $\theta$ in standard position is found

1. The problem is to clarify whether counting angles from $C270$ starts at 0 or 1. 2. In angle measurement, counting always starts at 0 degrees (or radians), representing the initi

1. The problem is about understanding how to count angles when dealing with transformations of the sine function, specifically for the function $-2\sin x$ and counting from $270^\c

1. Problem. Identify the transformations applied to the parent sine function to obtain $f(x) = -2\sin x$.

1. Problem. Identify the transformations applied to the parent sine function to obtain $f(x) = -2\sin x$.

1. **State the problem:** Solve the equation $$4 \sin \theta \cos \theta + \cos^2 \theta = 2 - \sin \theta$$ for $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Recall identities and

1. The problem is to understand and work with a trigonometric function. 2. Trigonometric functions relate angles to ratios of sides in right triangles and are periodic functions.