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1.3 Trigonometric functions  (Page 2/9)

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We can see that for a point P = ( x , y ) on a circle of radius r with a corresponding angle θ , the coordinates x and y satisfy

cos θ = x r x = r cos θ
sin θ = y r y = r sin θ .

The values of the other trigonometric functions can be expressed in terms of x , y , and r ( [link] ).

An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one blue line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another blue line segment that extends diagonally upwards and to the right to another point on the edge of the circle. This point is labeled “P = (x, y)”. These line segments have a length of “r” units. Between these line segments within the circle is the label “theta”, representing the angle between the segments. From the point “P”, there is a blue vertical line that extends downwards until it hits the x axis and thus hits the horizontal line segment, at a point labeled “x”. At the intersection horizontal line segment and vertical line segment at the point x, there is a right triangle symbol. From the point “P”, there is a dotted horizontal line segment that extends left until it hits the y axis at a point labeled “y”.
For a point P = ( x , y ) on a circle of radius r , the coordinates x and y satisfy x = r cos θ and y = r sin θ .

[link] shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of sin θ and cos θ .

Values of sin θ And cos θ At major angles θ In the first quadrant
θ s i n θ c o s θ
0 0 1
π 6 1 2 3 2
π 4 2 2 2 2
π 3 3 2 1 2
π 2 1 0

Evaluating trigonometric functions

Evaluate each of the following expressions.

  1. sin ( 2 π 3 )
  2. cos ( − 5 π 6 )
  3. tan ( 15 π 4 )
  1. On the unit circle, the angle θ = 2 π 3 corresponds to the point ( − 1 2 , 3 2 ) . Therefore, sin ( 2 π 3 ) = y = 3 2 .
    An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment that extends diagonally upwards and to the left to another point on the edge of the circle. This point is labeled “(-(1/2), ((square root of 3)/2))”. These line segments have a length of 1 unit. From the point “(-(1/2), ((square root of 3)/2))”, there is a vertical line that extends downwards until it hits the x axis. Inside the circle, there is a curved arrow that starts at the horizontal line segment and travels counterclockwise until it hits the diagonal line segment. This arrow has the label “theta = (2 pi)/3”.
  2. An angle θ = − 5 π 6 corresponds to a revolution in the negative direction, as shown. Therefore, cos ( − 5 π 6 ) = x = − 3 2 .
    An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment that extends diagonally downwards and to the left to another point on the edge of the circle. This point is labeled “(-((square root of 3)/2)), -(1/2))”. These line segments have a length of 1 unit. From the point “(-((square root of 3)/2)), -(1/2))”, there is a vertical line that extends upwards until it hits the x axis. Inside the circle, there is a curved arrow that starts at the horizontal line segment and travels clockwise until it hits the diagonal line segment. This arrow has the label “theta = -(5 pi)/6”.
  3. An angle θ = 15 π 4 = 2 π + 7 π 4 . Therefore, this angle corresponds to more than one revolution, as shown. Knowing the fact that an angle of 7 π 4 corresponds to the point ( 2 2 , − 2 2 ) , we can conclude that tan ( 15 π 4 ) = y x = −1 .
    An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment that extends diagonally downwards and to the right to another point on the edge of the circle. This point is labeled “(((square root of 2)/2), -((square root of 2)/2))”. These line segments have a length of 1 unit. From the point “(((square root of 2)/2), -((square root of 2)/2))”, there is a vertical line that extends upwards until it hits the x axis and thus the horizontal line segment. Inside the circle, there is a curved arrow that starts at the horizontal line segment and travels counterclockwise. The arrow makes one full rotation around the circle and then keeps traveling until it hits the diagonal line segment. This arrow has the label “theta = (15 pi)/4”.
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Evaluate cos ( 3 π / 4 ) and sin ( − π / 6 ) .

cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2

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As mentioned earlier, the ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at either of the acute angles of the triangle. Let θ be one of the acute angles. Let A be the length of the adjacent leg, O be the length of the opposite leg, and H be the length of the hypotenuse. By inscribing the triangle into a circle of radius H , as shown in [link] , we see that A , H , and O satisfy the following relationships with θ :

sin θ = O H csc θ = H O cos θ = A H sec θ = H A tan θ = O A cot θ = A O
An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment with length labeled “H” that extends diagonally upwards and to the right to another point on the edge of the circle. From the point, there is vertical line with a length labeled “O” that extends downwards until it hits the x axis and thus the horizontal line segment at a point with a right triangle symbol. The distance from this point to the center of the circle is labeled “A”. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta”.
By inscribing a right triangle in a circle, we can express the ratios of the side lengths in terms of the trigonometric functions evaluated at θ .

Constructing a wooden ramp

A wooden ramp is to be built with one end on the ground and the other end at the top of a short staircase. If the top of the staircase is 4 ft from the ground and the angle between the ground and the ramp is to be 10 ° , how long does the ramp need to be?

Let x denote the length of the ramp. In the following image, we see that x needs to satisfy the equation sin ( 10 ° ) = 4 / x . Solving this equation for x , we see that x = 4 / sin ( 10 ° ) ≈ 23.035 ft.

An image of a ramp and a staircase. The ramp starts at a point and increases diagonally upwards and to the right at an angle of 10 degrees for x feet. At the end of the ramp, which is 4 feet off the ground, a staircase descends downwards and to the right.
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A house painter wants to lean a 20 -ft ladder against a house. If the angle between the base of the ladder and the ground is to be 60 ° , how far from the house should she place the base of the ladder?

10 ft

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Trigonometric identities

A trigonometric identity    is an equation involving trigonometric functions that is true for all angles θ for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.

Rule: trigonometric identities

Reciprocal identities

tan θ = sin θ cos θ cot θ = cos θ sin θ csc θ = 1 sin θ sec θ = 1 cos θ

Pythagorean identities

sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ

Addition and subtraction formulas

sin ( α ± β ) = sin α cos β ± cos α sin β
cos ( α ± β ) = cos α cos β ∓ sin α sin β

Double-angle formulas

sin ( 2 θ ) = 2 sin θ cos θ
cos ( 2 θ ) = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ = cos 2 θ − sin 2 θ
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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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