Trigonometric Functions and Rules Questionbank

A student attempts to solve a triangle $LMN$ where side $l = 5$, side $m = 12$, and angle $L = 30^\circ$. How many distinct triangles can be formed with these specific dimensions?

A student attempts to find an angle $A$ in a triangle with side lengths $a = 20, b = 10,$ and $c = 8$. Their calculator returns a 'domain error'. What is the most likely mathematical reason for this?

A triangle has side lengths of $8$, $11$, and $15$. Calculate the measure of the smallest angle in the triangle to one decimal place.

A sinusoidal graph of the form $y = a \cos(b(x - h)) + k$, where $a, b > 0$, has its first maximum point at $(25^\circ, 12)$ and the immediately following point where the graph intersects its midline $y = k$ at $(55^\circ, 7)$. What are the amplitude and the period of this function?

During the conceptual justification of the sine rule, a perpendicular height $h$ is dropped from vertex $A$ to the opposite side $a$. Which pair of expressions correctly represents the height $h$ using right-triangle trigonometry?

To derive the sine rule for angles $A$ and $C$ in triangle $ABC$, an altitude $h$ is dropped from vertex $B$ to side $b$. Which equation correctly expresses $h$ in two different ways using right-triangle trigonometry?

A triangle has side lengths $a=4$, $b=5$, and $c=8$. If you calculate $\cos C$ using the cosine rule, what does the resulting value indicate about angle $C$?

Determine the value of the expression $\sin(90^\circ) - \cos(180^\circ)$.

Fill in the Blank: A Ferris wheel completes one full rotation every 90 seconds. To model the height in degrees using $h(t) = A\cos(Bt) + D$ where $t$ is in seconds, the value of the parameter $B$ is ________.

The tide in a harbor reaches a maximum height of $5\text{ m}$ at $t = 0$ hours and a minimum height of $1\text{ m}$ at $t = 6$ hours. Which of the following equations best models the height $h(t)$ for $t \ge 0$?