7.4 The other trigonometric functions (Page 3/14)

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Use reference angles to find all six trigonometric functions of $- \frac{7 π}{4} .$

$\begin{array}{l} \sin (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \cos (\frac{- 7 π}{4}) = \frac{\sqrt{2}}{2}, \tan (\frac{- 7 π}{4}) = 1, \\ \sec (\frac{- 7 π}{4}) = \sqrt{2}, \csc (\frac{- 7 π}{4}) = \sqrt{2}, \cot (\frac{- 7 π}{4}) = 1 \end{array}$

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Using even and odd trigonometric functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Consider the function $f (x) = x^{2},$ shown in [link] . The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. This matches the result of calculation: ${(4)}^{2} = {(−4)}^{2}, {(−5)}^{2} = {(5)}^{2},$ and so on. So $f (x) = x^{2}$ is an even function, a function such that two inputs that are opposites have the same output. That means $f (- x) = f (x) .$

This is an image of a graph of and upward facing parabola with points (-2, 4) and (2, 4) labeled. — The function $f (x) = x^{2}$ is an even function.

Now consider the function $f (x) = x^{3},$ shown in [link] . The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values. So $f (x) = x^{3}$ is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means $f (- x) = - f (x) .$

This is an image of a graph of the function f of x = x to the third power with labels for points (-1, -1) and (1, 1). — The function $f (x) = x^{3}$ is an odd function.

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in [link] . The sine of the positive angle is $y .$ The sine of the negative angle is $- y .$ The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in [link] .

Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.

$\begin{matrix} sin t & = & y \\ sin (- t) & = & - y \\ sin t & \neq & sin (- t) \end{matrix}$	$\begin{matrix} cos t & = & x \\ cos (- t) & = & x \\ cos t & = & cos (- t) \end{matrix}$	$\begin{matrix} tan (t) & = & \frac{y}{x} \\ tan (- t) & = & - \frac{y}{x} \\ tan t & \neq & tan (- t) \end{matrix}$
$\begin{matrix} sec t & = & \frac{1}{x} \\ sec (- t) & = & \frac{1}{x} \\ sec t & = & sec (- t) \end{matrix}$	$\begin{matrix} csc t & = & \frac{1}{y} \\ csc (- t) & = & \frac{1}{- y} \\ csc t & \neq & csc (- t) \end{matrix}$	$\begin{matrix} cot t & = & \frac{x}{y} \\ cot (- t) & = & \frac{x}{- y} \\ cot t & \neq & cot (- t) \end{matrix}$

A General Note

Even and odd trigonometric functions

An even function is one in which $f (- x) = f (x) .$

An odd function is one in which $f (- x) = - f (x) .$

Cosine and secant are even:

\begin{matrix} \cos (- t) & = & cos t \\ \sec (- t) & = & sec t \end{matrix}

Sine, tangent, cosecant, and cotangent are odd:

\begin{matrix} sin (- t) & = & - sin t \\ tan (- t) & = & - tan t \\ csc (- t) & = & - csc t \\ cot (- t) & = & - cot t \end{matrix}

Using even and odd properties of trigonometric functions

If the secant of angle $t$ is 2, what is the secant of $- t ?$

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle $t$ is 2, the secant of $- t$ is also 2.

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If the cotangent of angle $t$ is $\sqrt{3},$ what is the cotangent of $- t ?$

$- \sqrt{3}$

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Recognizing and using fundamental identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

A General Noter

Fundamental identities

We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

\tan t = \frac{\sin t}{\cos t}

\sec t = \frac{1}{\cos t}

\csc t = \frac{1}{\sin t}

cot t = \frac{1}{tan t} = \frac{cos t}{sin t}

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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