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How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of 5 π 12 with the ground? Round to the nearest foot.

About 52 ft

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Key equations

Trigonometric Functions Sine sin  t = opposite hypotenuse Cosine cos  t = adjacent hypotenuse Tangent tan  t = opposite adjacent Secant sec  t = hypotenuse adjacent Cosecant csc  t = hypotenuse opposite Cotangent cot  t = adjacent opposite
Reciprocal Trigonometric Functions sin  t = 1 csc  t csc  t = 1 sin  t cos  t = 1 sec  t sec  t = 1 cos  t tan  t = 1 cot  t cot  t = 1 tan  t
Cofunction Identities cos  t = sin ( π 2 t ) sin  t = cos ( π 2 t ) tan  t = cot ( π 2 t ) cot  t = tan ( π 2 t ) sec  t = csc ( π 2 t )

Key concepts

  • We can define trigonometric functions as ratios of the side lengths of a right triangle. See [link] .
  • The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See [link] .
  • We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See [link] .
  • Any two complementary angles could be the two acute angles of a right triangle.
  • If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See [link] .
  • We can use trigonometric functions of an angle to find unknown side lengths.
  • Select the trigonometric function representing the ratio of the unknown side to the known side. See [link] .
  • Right-triangle trigonometry facilitates the measurement of inaccessible heights and distances.
  • The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See [link] .

Section exercises

Verbal

For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.


A right triangle.


A right triangle with side opposite, adjacent, and hypotenuse labeled.

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When a right triangle with a hypotenuse of 1 is placed in a circle of radius 1, which sides of the triangle correspond to the x - and y -coordinates?

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The tangent of an angle compares which sides of the right triangle?

The tangent of an angle is the ratio of the opposite side to the adjacent side.

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What is the relationship between the two acute angles in a right triangle?

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Explain the cofunction identity.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

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Algebraic

For the following exercises, use cofunctions of complementary angles.

cos ( 34° ) = sin ( ___° )

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cos ( π 3 ) = sin ( ___ )

π 6

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csc ( 21° ) = sec ( ___° )

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tan ( π 4 ) = cot ( ___ )

π 4

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For the following exercises, find the lengths of the missing sides if side a is opposite angle A , side b is opposite angle B , and side c is the hypotenuse.

cos B = 4 5 , a = 10

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sin B = 1 2 , a = 20

b = 20 3 3 , c = 40 3 3

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tan A = 5 12 , b = 6

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tan A = 100 , b = 100

a = 10,000 , c = 10,00.5

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Questions & Answers

sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
Practice Key Terms 6

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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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