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Algebraic

For the following exercises, find the exact value of each expression.

For the following exercises, use reference angles to evaluate the expression.

sec 7 π 6

2 3 3

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If sin t = 3 4 , and t is in quadrant II, find cos t , sec t , csc t , tan t , cot t .

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If cos t = 1 3 , and t is in quadrant III, find sin t , sec t , csc t , tan t , cot t .

If sin t = 2 2 3 , sec t = 3 , csc t = 3 2 4 , tan t = 2 2 , cot t = 2 4

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If tan t = 12 5 , and 0 t < π 2 , find sin t , cos t , sec t , csc t , and cot t .

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If sin t = 3 2 and cos t = 1 2 , find sec t , csc t , tan t , and cot t .

sec t = 2 , csc t = 2 3 3 , tan t = 3 , cot t = 3 3

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If sin 40° 0.643 cos 40° 0.766 sec 40° , csc 40° , tan 40° , and cot 40° .

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If sin t = 2 2 , what is the sin ( t ) ?

2 2

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If cos t = 1 2 , what is the cos ( t ) ?

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If sec t = 3.1 , what is the sec ( t ) ?

3.1

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If csc t = 0.34 , what is the csc ( t ) ?

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If tan t = 1.4 , what is the tan ( t ) ?

1.4

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If cot t = 9.23 , what is the cot ( t ) ?

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Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 2 2 , cos t = 2 2 , tan t = 1 , cot t = 1 , sec t = 2 , csc t = 2

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Graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 3 2 , cos t = 1 2 , tan t = 3 , cot t = 3 3 , sec t = 2 , csc t = 2 3 3

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Technology

For the following exercises, use a graphing calculator to evaluate.

Extensions

For the following exercises, use identities to evaluate the expression.

If tan ( t ) 2.7 , and sin ( t ) 0.94 , find cos ( t ) .

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If tan ( t ) 1.3 , and cos ( t ) 0.61 , find sin ( t ) .

sin ( t ) 0.79

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If csc ( t ) 3.2 , and cos ( t ) 0.95 , find tan ( t ) .

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If cot ( t ) 0.58 , and cos ( t ) 0.5 , find csc ( t ) .

csc t 1.16

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Determine whether the function f ( x ) = 2 sin x cos x is even, odd, or neither.

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Determine whether the function f ( x ) = 3 sin 2 x cos x + sec x is even, odd, or neither.

even

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Determine whether the function f ( x ) = sin x 2 cos 2 x is even, odd, or neither.

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Determine whether the function f ( x ) = csc 2 x + sec x is even, odd, or neither.

even

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For the following exercises, use identities to simplify the expression.

sec t csc t

sin t cos t = tan t

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Real-world applications

The amount of sunlight in a certain city can be modeled by the function h = 15 cos ( 1 600 d ) , where h represents the hours of sunlight, and d is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42 nd day of the year. State the period of the function.

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The amount of sunlight in a certain city can be modeled by the function h = 16 cos ( 1 500 d ) , where h represents the hours of sunlight, and d is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267 th day of the year. State the period of the function.

13.77 hours, period: 1000 π

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The equation P = 20 sin ( 2 π t ) + 100 models the blood pressure, P , where t represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

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The height of a piston, h , in inches, can be modeled by the equation y = 2 cos x + 6 , where x represents the crank angle. Find the height of the piston when the crank angle is 55° .

7.73 inches

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The height of a piston, h , in inches, can be modeled by the equation y = 2 cos x + 5 , where x represents the crank angle. Find the height of the piston when the crank angle is 55° .

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Questions & Answers

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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