HPC Ch1 practice

(Calculator) Solve the triangle by finding all remaining side lengths and angle measures. If there is no triangle that satisfies the given conditions, type “0” in each blank. If there is more than one triangle that is consistent with the given conditions, write the triangle that satisfies [math]c[/math] is the longest side.
Note: round your answers to nearest integers.

(a) [math]A=30^{\circ}, b=12, a=10[/math]

(b) [math]B=125^{\circ}, b=10, c=12[/math]

(Calculator) A fugitive in a car is travelling across the Bonneville Salt Flats (of land speed records fame) in Utah. He leaves the road at an angle of [math]N60 ^{\circ}W[/math] and drives straight in a straight line at [math]140 km/h [/math]. At the moment he leaves the road, the police officer is pursuing him is exactly 100 m behind him. She leaves the road at an angle of [math]N45^{\circ}W[/math] intending to intercept his car. How far will each car travel to reach the point where their two paths meets? (1 decimal place)

(Calculator) Given the measurements below, determine how many triangles are possible.

(a) [math]A=40^{\circ}, a=15, c=12[/math]

(b) [math]B=55^{\circ}, b=10, a=12[/math]

(c) [math]a=3, b=4, c=8[/math]

(d) [math]A=114^{\circ}, a=11, c=8[/math]

(Calculator) Two ships leave a harbor, one travelling at 40 km/hr and the other at 35 km/hr. If the two ships are traveling on courses that have an angle of [math]140 ^{\circ}[/math] between them, how far apart are the ships (in km) after 6 hours. Round to nearest integer.

If [math]\sin 135^{\circ}=\sin x[/math], then [math]x[/math] could equal

If [math]\cos 60^{\circ}=\cos n[/math], then [math]n[/math] could equal

If [math]0^{\circ}\leq\theta\leq360^{\circ}[/math] and [math](\sin \theta)(\cos \theta)<0[/math], which of the following gives the possible value of [math]\theta[/math]?

Find [math]\sin {\theta}[/math] and [math]\cos {\theta}[/math] if [math]cot{\theta}=\dfrac{8}{15}[/math] and [math]\sec {\theta}<0[/math]

If [math]\sin x = m[/math] and [math]0^{\circ}<\theta<90^{\circ}[/math], then [math]\tan x =[/math]

Given the angle [math]\theta=\dfrac{7\pi}{4}[/math]

(a) Convert to degree.

(b) Find the reference angle, [math]\theta'[/math] in degrees.

(c) Find the value of [math]\sec\theta\csc\theta[/math]

(d) Find an equation that gives all the angles (in radian) coterminal with [math]\theta[/math]

Given [math]\tan\theta=\dfrac{1}{\sqrt{3}}[/math], find [math]\csc\theta[/math], where [math]\pi\leq\theta\leq2\pi[/math].

Find [math]\theta[/math] in radians for all [math]0\leq\theta\leq2\pi[/math]. If you have two or more answers, write the biggest one.

(a) [math]\sec\theta=-\sqrt{2}[/math]

(b) [math]\cot\theta=0[/math]

Find the [math]x[/math]-coordinate such that point [math](x,y)[/math] on the unit circle that corresponds to the real number [math]\theta=-\dfrac{2\pi}{3}[/math]

Find the value of [math]8\sin\left(\dfrac{\pi}{3}\right)\cos\left(-\dfrac{7\pi}{6}\right)[/math].

Consider the following numbers:
[math]a=\sec\left(\dfrac{5\pi}{3}\right)[/math]
[math]b=\tan\left(\dfrac{5\pi}{3}\right)[/math]
[math]c=\csc\left(\dfrac{-\pi}{2}\right)[/math]
[math]d=\sin\left(\dfrac{13\pi}{6}\right)[/math]
[math]e=\cot\left(\dfrac{13\pi}{6}\right)[/math]
[math]f=\cos\left(\dfrac{-\pi}{4}-2000\pi\right)[/math]
How many numbers are negative and irrational?

Which of the following angles is not co-terminal to the others?

Convert:

(a) Degree [math]=[/math] [math]-40^\circ[/math]

(b) Radian [math]=[/math] [math]\dfrac{11}{9}\pi[/math]

Find the perimeter of the sector which radius is 7 and the central angle is [math]\dfrac{\pi}{4}[/math].

(Calculator) A wheel is rotating at 7 radians/sec, and the wheel has a 33-inch diameter. To the nearst foot per minute, what is the velocity of the point in the rim?

If the graph of [math]y=f(x)[/math] is shown, then which of the following could be true?

Determine a possible sine equation for the function [math]y=f(x)[/math] graphed below.

Write the equation for the graph.

Given the graph below, determine the equation of the graph.

Given the graph below, determine the equation of the graph.

The Ferris wheel shown in this figure makes one complete turn every 20 seconds. A rider’s height, h, above the ground can be modeled by the equation [math]h=a \sin \omega t+k[/math], where [math]h[/math] and [math]k[/math] are given in feet and [math]t[/math] is given in seconds.

(a) What is the value of [math]a[/math]?

(b) What is the value of [math]k[/math]?

(c) What is the value of [math]\omega[/math]?

An object in simple harmonic motion has a frequency of [math]\dfrac{1}{4}[/math] oscillation per minute and an amplitude of 8 feet. Write an equation in the form [math]d= \alpha \sin( \omega t )[/math] for the object’s simple harmonic motion.

If [math]\tan\dfrac{\pi}{8}=a+\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers, find the value of [math]a-b[/math].

If [math]\tan \theta+\cot \theta=5[/math], what is the value of [math]\csc^2\theta+\sec^2\theta [/math]?

Simplify [math]\sec x-\sin x\tan x[/math] into a single trigonmetric expression.

What is [math]\sin(8x)\cos(2x)+\cos(8x)\sin(2x)[/math]?

What is [math]\dfrac{\tan 37^{\circ}-\tan 13^{\circ}}{1+(\tan 37^{\circ})(\tan 13^{\circ})}[/math]?

Which one of the following trigonometric expressions is identical to [math]\cos x (\sec x-\cos x)[/math]?

When [math]\cos\theta=-\dfrac{5}{13}[/math], what is the value of [math]\cos (2\theta)[/math]?

Consider the equation [math]2\sin^2{x}-2\sin{x}+4\sin^3{x}=1[/math] where [math]x\in[0,2\pi][/math]

(a) How many solutions does this equation have?

(b) What is the maximum solution?

Solve the equation [math]2\cos\left(\dfrac{1}{2}x\right)\cos x-2\sin\left(\dfrac{1}{2}x\right)\sin x=1[/math]. State all real solutions using the exact form.

Solve the equation [math]\sin{(2\theta)}+\cos{(2\theta)}=0[/math] for all such that [math]0\leq\theta\leq2\pi[/math]. Write down your answers in ascending order.

Solve [math]\sqrt{3}\tan (2x)=-3[/math] for [math][0,\pi)[/math]. Write your answer in ascending order.

Find the solutions for [math]2\sqrt{3}\cot(3x)=6[/math] over the interval [math][0,2\pi)[/math]. Write your answer in ascending order.

Solve [math]5 \sec^2 x-2\tan x -8=0[/math].

Find the exact value of the expression [math]\tan\left(2\cos^{-1}\left(-\dfrac{3}{5}\right)\right)[/math].

Write the appropriate answer in the space provided.

(a) [math]\arcsin\left(\dfrac{\sqrt{2}}{2}\right)[/math]

(b) [math]\arccos\left(-\dfrac{\sqrt{2}}{2}\right)[/math]

(c) [math]\arctan\left(\sqrt{3}\right)[/math]

(d) [math]\sin^{-1}\left(\sin(-\pi)\right)[/math]

(e) [math]\tan^{-1}\left(\tan\dfrac{\pi}{6}\right)[/math]

(f) [math]\cos^{-1}\left(\cos\left(-\dfrac{5\pi}{6}\right)\right)[/math]

What is [math]\cos^{-1}\left(\cos\dfrac{5\pi}{4}\right)[/math]?

Simplify [math]\csc\left(\arccos\dfrac{x}{5}\right)[/math].

If [math]\sin\left(\tan^{-1}\dfrac{1}{2}-\cos^{-1}\dfrac{1}{4}\right)=\dfrac{b\sqrt{5}-c\sqrt{3}}{a}[/math] where [math]a, b[/math], and [math]c[/math] are positive integer, what is the values of [math]a, b[/math], and [math]c[/math]?

Simplify [math]\cot\left(\sin^{-1}\dfrac{3x}{5}\right)[/math].

Find the exact of the following expression [math]\tan\left(\arcsin\dfrac{4}{5}+\arccos1\right)[/math].

Find the exact value of the following expression [math]\csc\left(2\arcsin\left(-\dfrac{3}{5}\right)\right)[/math].

Consider the equation [math]4\cos3x+2=0, -\pi\leq x\leq\pi[/math].

(a) How many solutions does this equation have?

(b) What is the minimum solution?

Find the solutions of the equation [math]2\sin(3\theta)-1=0[/math] in the interval [math][0,2\pi)[/math]. Write your answer in ascending order.

Solve [math]x[/math] in [math][0,2\pi)[/math]: [math]\cos(x)-2\sin^2(x)=1[/math]. If you have two or more answers, write the biggest one.

(Calculator) Find the angle [math]Q[/math] of [math]\Delta PQR[/math] if [math]p=7, q=10[/math], and [math]r=5[/math]. (one decimal place)

(Calculator) Find the area of quadrilateral [math]ABCD[/math].

(Calculator) A plane flies a straight course. On the ground directly below the flight path, observes 2 miles apart spot the plane at the same time. The plane’s angle of elevation is [math]46^\circ[/math] from one observation point and [math]71^\circ[/math] from the other. How high is the plane?

(Calculator) Consider a triangle [math]ABC[/math] where [math]a=8[/math]cm, [math]b=14[/math]cm, and [math]c=10.2[/math]cm. Round your answer to nearest hundredth.

(a) Find [math]m\angle A, m\angle B[/math] and [math]m\angle C[/math].

(b) Find the area, in [math]\text{cm}^2[/math], of triangle [math]ABC[/math] using Heron’s formula.

(Calculator) Consider a triangle [math]ABC[/math] with [math]m\angle A=67^\circ[/math], [math]a=17[/math]cm and [math]b=14[/math]cm. Find [math]c[/math]. If you have two answers, write the biggest one.

(Calculator) Consider a triangle [math]ABC[/math] with [math]m\angle A=47^\circ[/math], [math]a=14[/math]cm and [math]b=17[/math]cm. Find [math]c[math]. Round your answer to nearest hundredth. If you have two answers, write the smallest one.

(Calculator) Calculate the area of the triangle shown. Round your answer to nearest hundredth.

(Calculator) Determine how many triangles can be created using the given information:

(a) [math]A=150^\circ, b=12, a=8[/math]

(b) [math]a=4, b=6, c=9[/math]

(c) [math]C=40^\circ, a=10, c=13[/math]

(d) [math]B=52^\circ, A=70^\circ, c=9[/math]

Given a triangle with [math]A=30^\circ[/math] and [math]b=12[/math] determine the restrictions on [math]a[/math] in order to form two angles.

(Calculator) Consider a triangle [math]ABC[/math] with [math]m\angle B=25^\circ[/math], [math]c=10[/math]cm and [math]b=6[/math]cm. Find [math]a[/math]. Round your answer to nearest hundredth. If you have two answers, write the biggest one.

(Calculator) If [math]\Delta ABC[/math] that satisfies [math]m\angle A=40^\circ[/math], [math]c=32[/math]cm, and [math]b=21[/math]cm. Find the largest angle. Round your answer to nearest hundredth.

Given the angle [math]\theta=-\dfrac{5\pi}{6}[/math]

(a) Convert to degree [math]0^{\circ} \leq \theta \leq 360^{\circ}[/math].

(b) Find the reference angle [math]\theta'[/math].

(c) Find the values of [math]\csc\theta[/math]

Let [math]\cos\theta=\dfrac{1}{3}[/math] where [math]\pi\leq\theta\leq 2\pi[/math]. If [math]\cot\theta=a\sqrt{2}[/math] where [math]a[/math] is rational, find [math]a[/math]

Find [math]\theta[/math] in radians for all [math]0\leq\theta\leq2\pi[/math]. Write your answer in ascending order.

(a) [math]\csc\theta=-\dfrac{2\sqrt{3}}{3}[/math]

(b) [math]\cot\theta= 1[/math]

(Calculator) A radial arm saw has a circular cutting blade with a diameter of 10 inches. It spins at 1800 rpm. If there are 14 cutting teeth per inch on the cutting blade, how many teeth cross the cutting surface each second? Round to nearest integer.

The point [math](-9,12)[/math] is on the terminal sided of [math]\theta[/math]. Find [math]\sin\theta[/math] and [math]\cos\theta[/math].

Let [math]\tan\theta=-\dfrac{2}{3}[/math] and [math]\sin\theta>0[/math]. If [math]\cos\theta=\dfrac{a}{\sqrt{b}}[/math]. where [math]a, b\in\mathbb{Z}[/math], what is the value of [math]a+b[/math]?

Which of the following expressions does not represent a real numbers?

Which one of the following is a positive value when the point [math]P(4,-5)[/math] is on the terminal side of the angle [math]\theta[/math] in standard position?

Let [math]\tan\theta=\dfrac{\sqrt{2}}{3}[/math] and [math]\sin\theta>0[/math]. If [math]\cos\theta=\dfrac{a}{\sqrt{b}}[/math]. where [math]a, b\in\mathbb{Z}[/math], what is the value of [math]a+b[/math]?

The radius of a wheel rolling on the ground is 60cm. If the wheel rotates through an angle of [math]60^\circ[/math], how many cm does it move?

A Ferris wheel has a radius 30 meters. The wheel is rotating at 2 revolutions per minute. Find the linear speed, in meters per minute, of a seat on this Ferris wheel.

If [math]f(x)=\sin x[/math] and [math]f(a)=\dfrac{1}{3}[/math], and [math]g(x)=\tan x[/math] and [math]g(b)=\dfrac{1}{2}[/math], find the value of

$$f(a-2\pi)+f(a+6\pi)-g(b)+g(b+3\pi)+g(b-2\pi)$$

Find the refernece angles of [math]2.8[/math]

Write the letter (A-U) that matches each expression.

(a) [math]\sec\left(\dfrac{-7\pi}{4}\right)[/math]

(b) [math]\cos\left(\dfrac{5\pi}{3}\right)[/math]

(c) [math]\csc^2\left(\dfrac{5\pi}{6}\right)-\cot^2\left(\dfrac{5\pi}{6}\right)[/math]

(d) [math]-\sin\left(\dfrac{\pi}{3}+2\pi\right)[/math]

(e) [math]\cot\left(\dfrac{\pi}{2}-\dfrac{\pi}{6}\right)[/math]

(f) [math]\csc\left(\dfrac{3\pi}{2}+\pi\right)[/math]

How many degrees is [math]\dfrac{14\pi}{5}[/math] radians?

What is the reference angle for [math]-\dfrac{21\pi}{5}[/math]?

Simplify the expression: [math]\sin\left(\dfrac{\pi}{2}+x\right)+\cos(\pi-x)[/math].

Solve for [math]\theta[/math] on the interval [math]0\leq\theta<2\pi[/math], when [math]\csc\theta=2[/math].

What is the period of the function [math]f(x)=100\cos(200x-300)+400[/math]?

The number of solutions to the equation [math]\sin(1000x)=\dfrac{5}{7}[/math] in the interval [math][0,2\pi][/math] is

The graph of [math]f(x)=\tan x[/math] can be obtained by a horizontal shift of the graph of

The graph of the trigonometric equation that can be written in the form [math]f(x)=A\cos(B(x-C))+D[/math] is shown. Find the value of [math]A, B, C[/math], and [math]D[/math].

Given the graph below, determine the equation of the graph.

Find one possible sine equation and one possible cosine equation for the graph shown. Each blank contains non-negative rational number.

A Ferris wheel with a diameter of 30 meters positioned 6 meters above the ground makes one complete turn every 30 minutes. Which equation correctly models the height of a rider who begins to ride at the bottom of the wheel when [math]t=0[math]?

Tides go up and down in a 18-hour period. The average depth of a certain river is 15 meters and ranges from 11 to 19 meters. Write a sine equation that gives the approximate variation [math]y[/math], if [math]x[/math] is the hours, after midnight and high tide occurs at 10:00 am.

If [math]\tan\left(\dfrac{\pi}{4}-\dfrac{\pi}{3}\right)=a+\sqrt{b}[/math] where [math]a, b[/math] are integers, what is [math]a+b[/math]?

If [math]\left(\sin67.5^\circ-\cos67.5^\circ\right)^2=a+b\sqrt{2}[/math] where [math]a, b[/math] are integers, what is [math]a+b[/math]?

If [math]f(x)=\sin x[/math] and [math]g(x)=\cos x[/math], then [math]g(2x)[/math]

Simplify [math]\sin(x+y)\cos(-x)-\cos(x+y)\sin(-x)[/math]

Let [math]\cos x=-\dfrac{\sqrt{7}}{4}[/math] where [math]\pi<x<2\pi[/math].If [math]\tan2x=a\sqrt{7}[/math] where [math]a[/math] is rational, what is [math]a[/math]?

If [math]\tan x+\tan y=25[/math] and [math]\cot x+\cot y=\dfrac{1}{6}[/math]. Find [math]\tan(x+y)[/math].

Solve [math]\dfrac{1}{4}(2\cos(2x)+\sec(2x))=\dfrac{3}{4}[/math] on the interval [math]-\pi\leq x\leq0[/math]. Write your answer in accending order.

Solve [math]\cot\theta\cos\theta+4\cot\theta=-3\cot\theta[/math] on the interval [math]0\leq\theta\leq2\pi[/math]. Write your answer in accending order.

Solve [math]\cos(2\theta)+3\cos\theta+2=0[/math].

How many solutions does the equation [math]\sin2x=\tan x[/math] have where [math]x\in[-\pi,\pi][/math]?

What is the range of the function [math]y=\cos(\tan^{-1}x)[/math]

Which of the following statement(s) is/are true?

Find the exact value of [math]\sin\left(\cos^{-1}\dfrac{5}{13}-\sin^{-1}\dfrac{3}{5}\right)[/math].

Evaluate the following numbers. If your answer is undefined, write “undef”.

(a) [math]\cos^{-1}\left(\cos\dfrac{5\pi}{4}\right)[/math]

(b) [math]\tan\left(\sin^{-1}\left(-\dfrac{4}{5}\right)\right)[/math]

(c) [math]\sin^{-1}\left(\dfrac{\pi}{2}\right)[/math]

(d) [math]\cos^{-1}\left(\sin\dfrac{4\pi}{5}\right)[/math]

Find all solutions to the equation [math]2\cos^2x=1[/math].

Find the summation of the all solutions to the equation [math]4\cos x\cos3x+2\cos x-6\cos3x-3=0[/math] where [math]-\pi\leq x\leq\pi[/math].

(Calculator) Let [math]ABC[/math] be a triangle satisfying [math]\angle A=80^\circ, a=23[/math], and [math]b=5[/math]. Find the other angles and side. If you have two triangles, write the obtuse one.

(Calculator) Let [math]ABC[/math] be a triangle satisfying [math]\angle A=19^\circ, b=7[/math], and [math]c=13[/math]. Find the other angles and sides. If you have two triangles, write the obtuse one.

(Calculator) Let [math]ABC[/math] be a triangle satisfying [math]\angle A=30^\circ, a=24[/math], and [math]b=42[/math]. Find the other angles and side. If you have two triangles, write the obtuse one.

If two angles (both acute) and one side are given , how many triangles are possible?

Jonathan uses the law of cosines correctly for a case in which [math]a, b, [/math] and [math]c[/math] are given. He decides to find [math]\angle A[/math] first. He gets a negative value of [math]\cos A[/math]. How many triangles are possible in this case?

(Calculator) Two air-planes leave an airport at the same time, one going [math]N45^\circ W[/math] at 400 mph and the other going directly east at 330 mph. How far apart are the planes after 4 hours (to the nearest mile)?

(Calculator) A boy standing 50 meters from the base of a building measures the angle to the top of the building and finds it to be [math]35^\circ[/math]. The boy then measures the angle to the top of the radio tower on the building and finds that it is [math]50^\circ[/math]. How tall, in meters, is the radio tower?

Find the area of [math]\Delta ABC[/math] if [math]a=8, b=15[/math] and [math]\angle C=40^\circ[/math].

If you were only allowed to ise the law of cosines, which of the following triangle(s) (given only these measurements) could be solved?

Let [math]ABC[/math] be a tirangle with lengths [math]AB=c, BC=a, AC=b[/math]. Which of the following value(s) is/are equal to [math]\dfrac{b}{a}[/math]?

(Calculator) The shortest side of a triangle with angles [math]40^\circ, 60^\circ[/math], and [math]80^\circ[/math] has a length of [math]9[/math]. What is the length of the longest side?

(Calculator) A tower is 100 feet tall. At a particular time of day, the tower cast a 150-foot-long shadow. What is the angle of elevation from the end of the shadow to the top of the tower?

(Calculator) Determine the values of [math]b[/math] that will produce 2 triangles if [math]a=20[/math] and [math]\angle B=40^\circ[/math].

(Calculator) Consider triangle [math]ABC[/math] with [math]\angle A=0.5, AB=9[/math], and [math]BC=6[/math]. Find [math]AC[/math]. If you have two answers, write the biggest one.