Trigonometric Functions

The projectile is fired at an angle of 30° to the horizontal with an initial speed of $150$ meters per second . Find the range R and height.

The projectile is fired at an angle of 25° to the horizontal with an initial speed of $500$ meters per second. Find the range R and height H.

The projectile is fired at an angle of 50° to the horizontal with an initial speed of 200 feet per second.Find the range and height.

If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the function

$t\left(\theta \right)=\sqrt{\frac{2a}{g \sin \theta \cos \theta }}$

where a is the length (in feet) of the base and g  32 feet per second per second is the acceleration due to gravity. How

long does it take a block to slide down an inclined plane with base a = 10 feet when,

(a) $\theta = 30°$

(b) $\theta =45°$

(c) $\theta =60°$

Piston Engines In a certain piston engine, the distance x (in cm) from the center of the drive shaft to the head of the piston is given by the function.

$x\left(\theta \right)=\cos \theta +\sqrt{16+0·5 \cos \left(2\theta \right)}$

where $\theta$ is the angle between the crank and the path of the piston head. See the figure. Find $x$ when $\theta =30°$ and $\theta =45°$.

Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a 

distance of 1 mile from a paved road that parallels the ocean.

See the figure.

Sally can jog 8 miles per hour along the paved road, but only 3 miles per hour in the sand on the beach. Because of a river directly between the two houses, it is necessary to jog in the sand to the road, continue on the road, and then jog directly back in the sand to get from one house to the other. The time T to get from one house to the other as a function of the angle $\theta$  shown in the illustration.

$T\left(\theta \right)=1+\frac{2}{3 \sin \theta }-\frac{1}{4 \tan \theta }$and $\theta$ lies between $0$ to $90°$.

(a)  

Calculate the time T for $\theta =30°$. How long is Sally on the paved road?

(b)  

Calculate the time T for $\theta = 45°$. How long is Sally on the paved road?

(c) 

Calculate the time T for $\theta =60°$ How long is Sally on the paved road?

(d) 

Calculate the time T for $\theta =90°$. Describe the path taken. Why can't the formula for T be used?

Designing Fine Decorative Pieces A designer of decorative art plans to market solid gold spheres encased in clear crystal

cones. Each sphere is of fixed radius R and will be enclosed in a cone of height h and radius r. See the illustration.Many cones can be used to enclose the sphere, each having

a different slant angle u. The volume V of the cone can be expressed as a function of the slant angle $\theta$ of the cone as,

$V\left(\theta \right)=\frac{1}{3}\pi R^3\frac{{\left(1+sec\theta \right)}^3}{{\left(\tan \theta \right)}^3}$

where $\theta$ lies between $0°$ and $90°$.

What volume V is required to enclose a sphere of radius 2 cm in a cone whose slant angle $\theta$ is 30°? 45°? 60°?

Projectile Motion - An object is propelled upward at an angle $\theta$ between $45°$and $90°$ to the horizontal with an initial

velocity of ${\nu }_o$ feet per second from the base of an inclined plane that makes an angle of 45° with the horizontal. See the

illustration. If air resistance is ignored, the distance R that it travels up the inclined plane as a function of $\theta$ is given by

$R\left(\theta \right)=\frac{{{\nu }_o}^2\sqrt{2}}{32}\left[\sin \left(2\theta \right)-\cos \left(2\theta \right)-1\right]$

(a) Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and $\theta =60°$

(b) Graph R = R$\left(\theta \right)$ if the initial velocity is 32 feet per second.

(c) What value of $\theta$ makes R largest?

If $\theta ,0<\theta <\mathrm{\pi }$, is the angle between the positive x-axis and a nonhorizontal, nonvertical line L, show that the slope m of L equals $\tan \theta$. The angle $\theta$ is called the inclination of L.

[Hint: See the illustration, where we have drawn the line M parallel to L and passing through the origin. Use the fact that M intersects the unit circle at the point $(\cos \theta , \sin \theta )$.]

In Problem$131$, use the figure to approximate the value of

the six trigonometric functions at t to the nearest tenth. Then use a

calculator to approximate each of the six trigonometric functions at t.

(a) $t=1$

(b) $t=5.1$

Use the figure to approximate the value of the six trigonometric functions at t to the nearest tenth. Then use a calculator to approximate each of the six trigonometric functions at t. 

a)$t=2$

b)$t=4$

Write a brief paragraph that explains how to quickly compute the trigonometric functions of 30°, 45°, and 60°.

Write a brief paragraph that explains how to quickly compute the trigonometric functions of 0°, 90°, 180°, and 270°.

How would you explain the meaning of the sine function to a fellow student who has just completed college algebra?

The domain of the function $f\left(x\right)=\frac{x+1}{2x+1}$ is _______.

A function for which $f\left(x\right)=f(-x)$ for all $x$ in the domain of $f$ is called a(n) ___ function.

True or false  The function $f\left(x\right)=\sqrt{x}$ is even.

The equation $x^2+2x={\left(x+1\right)}^2-1$ is an identity.

The domain of the tangent function is _____.

The range of the sine function is _____.

True or False  The only even trigonometric function are the cosine and secant functions.

$si{n}^2\theta +co{s}^2\theta =$____.

True or False  $sec\theta =\frac{1}{\sin \theta }$.

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sin 405°$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\cos \left(420°\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\tan 405°$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sin 390°$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\csc \left({450}^{\circ \hspace{0.17em}}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sec \left({540}^{\circ \hspace{0.17em}}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\cot \left({390}^{\circ \hspace{0.17em}}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sec \left({420}^{\circ \hspace{0.17em}}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\cos \left(\frac{33\pi }{4}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sin \left(\frac{9\pi }{4}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\tan \left(21\mathrm{\pi }\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\csc \left(\frac{9\pi }{2}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sec \left(\frac{7\pi }{4}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\cot \left(\frac{17\pi }{4}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\tan \left(\frac{19\pi }{6}\right)$

In problems 11-26, use the fact that the trigonometric function are periodic to find the exact value of each expression. Do not use a calculator.

$\sec \left(\frac{25\pi }{6}\right)$

In problems 27-34, name the quadrant in which the angle $\theta$ lies.

$\sin \theta >0, \cos \theta <0$

In problems 27-34, name the quadrant in which the angle $\theta$ lies.

$\sin \theta <0, \cos \theta >0$

In problems 27-34, name the quadrant in which the angle $\theta$ lies.

$\sin \theta <0, \tan \theta <0$

In problems 27-34, name the quadrant in which the angle $\theta$ lies.

$\cos \theta >0, \tan \theta >0$

Name the quadrant in which the angle $\theta$ lies.

$\cos \theta > 0, \tan \theta <0$

Name the quadrant in which the angle $\theta$ lies.

$\cos \theta >0, \tan \theta <0$

Name the quadrant in which the angle $\theta$ lies.

$\cos \theta <0,\tan \theta >0$

Name the quadrant in which the angle $\theta$ lies.

$sec<0, \sin \theta >0$

Name the quadrant in which the angle $\theta$ lies. $csc\theta >0,\cos \theta <0$

$\sin \theta ,\cos \theta$ are given. Find the exact value of each of the four remaining trigonometric functions.$\sin \theta =- \frac{3}{5} ,\cos \theta =\frac{4}{5}$

$\sin \theta and \cos \theta$ are given .Find the exact value of each of the four remaining trigonometric functions $\sin \theta = \frac{4}{5} ,\cos \theta =-\frac{3}{5}$