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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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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1+cos²A/cos²A=2cosec²A-1
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a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
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100 meters
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Find that number sum and product of all the divisors of 360
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answer
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exponential series
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Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
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10y
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prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
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why is {2kπ} union {kπ}={kπ}? when k belong to integer
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if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
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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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