📏 trigonometry

1. The problem is to simplify the expression $\cos^3 x + \cos^4 x$. 2. Factor out the common term $\cos^3 x$ from both parts:

1. The problem is to determine whether the function $f(x) = \cos^7(x)$ is odd or even. 2. Recall that a function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.

1. **Problem Statement:** We have $$f(x) = \sqrt{3}\cos x + \sin x$$

1. State the problem: Solve for x given $\cos(3x+6^\circ)=\tfrac{1}{2}$ and $3x+6^\circ$ is an acute angle. 2. Understand the acute-angle constraint: an acute angle means it lies s

1. **Problem statement:** Given $\sin a = \cos b$ where $a,b$ are acute angles, find $\tan(a+b)$. Since $a,b$ are acute and $\sin a = \cos b$, then $a = b$ because sine and cosine

1. Given that $\sin a = \cos b$ and $a, b$ are acute angles, recall that $\sin a = \cos (90^\circ - a)$, so $b = 90^\circ - a$. We want to find $\tan (a+b)$. 2. Substitute $b = 90^

1. Ex1 Problem: Find $x$ given $0^\circ < x < 90^\circ$ with various trigonometric equations. 1) Solve $\sin 2x = \cos 4x$.

1. The problem asks to find the length of side MK (opposite side) in a right triangle where the hypotenuse is 10 meters and the angle \( \theta = 30^\circ \). 2. We use the sine fu

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 +