📏 trigonometry

1. Problem 1: Find length AB in the triangle where opposite side $=8$ cm, angle $\theta = 60^\circ$, and adjacent side is AB.\n 2. Use the tangent function: $$\tan \theta = \frac{\

1. Problem: Determine the size of the angle BAC in a right triangle where the hypotenuse AC is 16 cm and the base BC is 8 cm. 2. In a right triangle, use cosine to find the angle b

1. **Problem 1: Ship's Journey** - The ship sails 125 km on a bearing of 080°.

1. The problem asks to evaluate $2.14 \sin 75^\circ$ and round the answer to 2 decimal places. 2. Recall that $\sin 75^\circ$ is the sine of 75 degrees.

1. **Problem statement:** Find the first day $t$ after the spring equinox where the day length $L(t)$ equals 750 minutes, given:

1. **State the problem:** Find the value of $\cos^{-1}(0.32)$.

1. The problem is to simplify the expression $\sin \left( \frac{9\pi}{8} \right) \cos \left( \frac{\pi}{8} \right) - \cos \left( \frac{9\pi}{8} \right) \sin \left( \frac{\pi}{8} \r

1. State the problem: Find the exact values of $\cos\left(\frac{3\pi}{4}\right)$ and $\sin\left(\frac{3\pi}{4}\right)$.\n\n2. Recall that $\frac{3\pi}{4}$ radians is in the second

1. The problem is to solve a triangle that is not a right triangle, given one side and one angle. 2. Since it is not a right triangle, we cannot use simple trigonometric ratios lik

1. The problem asks us to evaluate the expression $$d=\frac{h}{\tan(23.6^\circ)}$$ where $h$ is a variable and $23.6^\circ$ is the angle in degrees. 2. Recall that $$\tan(\theta)$$

1. The problem states that the angle of depression from the top of a building to a point P on the ground is 23.6°. 2. We want to find the horizontal distance from the foot of the b

1. Solve the following trig equations for $0 \leq \theta \leq 360^\circ$. **i)** $2 + 4 \cos^2 \theta = 7 \cos \theta \sin \theta$

1. Stating the problem: Given the equation $m \sin\theta = n \sin(\theta + 2)$, find the value of $\frac{m+n}{m-n} \tan \theta$. 2. Use the sine addition formula for $\sin(\theta +

1. The problem states: If $m \sin \theta = n \sin (\theta + 2)$, then find the value of $\frac{m+n}{m-n} \tan a$ in terms of tangent expressions involving $\theta$ and $a$. 2. Give

1. The problem states a triangle $PQR$ with angles $\alpha$, $\beta$, and $\gamma$ such that $\alpha + \beta + \gamma = \pi$. 2. We need to verify or simplify the expression: $\sin

1. The problem states that in a triangle with angles $\alpha$, $\beta$, and $\gamma$, we have $\alpha + \beta + \gamma = \pi$. 2. We need to verify or solve the expression $\sin^2

1. **Determine values of a and b for** $f(x)=a\cos(x+b)$ from the graph. - The amplitude $a$ is the maximum value of $f(x)$.

1. Let's clarify your problem: you want to understand the expression involving the power of $\tan \frac{x}{y}$ and why it might be wrong or misinterpreted. 2. The function $\tan \f

1. Let's analyze the expression $\frac{\sin(b/2) \sin(90+a)}{\cos(a/2) \sin(b)}$. 2. Recall that $\sin(90^\circ + x) = \cos(x)$ (using degrees). So, $\sin(90 + a) = \cos(a)$.

1. The problem is to express \(\cos 5x\) in terms of powers of \(\cos x\) using De Moivre's theorem. 2. De Moivre's theorem states that \((\cos x + i \sin x)^n = \cos nx + i \sin n

1. The problem is to find the value of $\sin 60^\circ$.\n\n2. We know from the properties of special right triangles that a 60-degree angle is present in an equilateral triangle sp