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Let us look back at our values for tan θ

θ 0 30 45 60 90 180
tan θ 0 1 3 1 3 0

Now that we have graphs for sin θ and cos θ , there is an easy way to visualise the tangent graph. Let us look back at our definitions of sin θ and cos θ for a right-angled triangle.

sin θ cos θ = opposite hypotenuse adjacent hypotenuse = opposite adjacent = tan θ

This is the first of an important set of equations called trigonometric identities . An identity is an equation, which holds true for any value which is put into it. In this case we have shown that

tan θ = sin θ cos θ

for any value of θ .

So we know that for values of θ for which sin θ = 0 , we must also have tan θ = 0 . Also, if cos θ = 0 our value of tan θ is undefined as we cannot divide by 0. The graph is shown in [link] . The dashed vertical lines are at the values of θ where tan θ is not defined.

The graph of tan θ .

Functions of the form y = a tan ( x ) + q

In the figure below is an example of a function of the form y = a tan ( x ) + q .

The graph of 2 tan θ + 1 .

Functions of the form y = a tan ( θ ) + q :

  1. On the same set of axes, plot the following graphs:
    1. a ( θ ) = tan θ - 2
    2. b ( θ ) = tan θ - 1
    3. c ( θ ) = tan θ
    4. d ( θ ) = tan θ + 1
    5. e ( θ ) = tan θ + 2
    Use your results to deduce the effect of q .
  2. On the same set of axes, plot the following graphs:
    1. f ( θ ) = - 2 · tan θ
    2. g ( θ ) = - 1 · tan θ
    3. h ( θ ) = 0 · tan θ
    4. j ( θ ) = 1 · tan θ
    5. k ( θ ) = 2 · tan θ
    Use your results to deduce the effect of a .

You should have found that the value of a affects the steepness of each of the branches. The larger the absolute magnitude of a , the quicker the branches approach their asymptotes, the values where they are not defined. Negative a values switch the direction of the branches. You should have also found that the value of q affects the vertical shift as for sin θ and cos θ . These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = a tan ( x ) + q .
a > 0 a < 0
q > 0
q < 0

Domain and range

The domain of f ( θ ) = a tan ( θ ) + q is all the values of θ such that cos θ is not equal to 0. We have already seen that when cos θ = 0 , tan θ = sin θ cos θ is undefined, as we have division by zero. We know that cos θ = 0 for all θ = 90 + 180 n , where n is an integer. So the domain of f ( θ ) = a tan ( θ ) + q is all values of θ , except the values θ = 90 + 180 n .

The range of f ( θ ) = a tan θ + q is { f ( θ ) : f ( θ ) ( - , ) } .

Intercepts

The y -intercept, y i n t , of f ( θ ) = a tan ( x ) + q is again simply the value of f ( θ ) at θ = 0 .

y i n t = f ( 0 ) = a tan ( 0 ) + q = a ( 0 ) + q = q

Asymptotes

As θ approaches 90 , tan θ approaches infinity. But as θ is undefined at 90 , θ can only approach 90 , but never equal it. Thus the tan θ curve gets closer and closer to the line θ = 90 , without ever touching it. Thus the line θ = 90 is an asymptote of tan θ . tan θ also has asymptotes at θ = 90 + 180 n , where n is an integer.

Graphs of trigonometric functions

  1. Using your knowldge of the effects of a and q , sketch each of the following graphs, without using a table of values, for θ [ 0 ; 360 ]
    1. y = 2 sin θ
    2. y = - 4 cos θ
    3. y = - 2 cos θ + 1
    4. y = sin θ - 3
    5. y = tan θ - 2
    6. y = 2 cos θ - 1
  2. Give the equations of each of the following graphs:

The following presentation summarises what you have learnt in this chapter. Ignore the last slide.

End of chapter exercises

  1. Calculate the unknown lengths
  2. In the triangle P Q R , P R = 20  cm, Q R = 22  cm and P R ^ Q = 30 . The perpendicular line from P to Q R intersects Q R at X . Calculate
    1. the length X R ,
    2. the length P X , and
    3. the angle Q P ^ X
  3. A ladder of length 15 m is resting against a wall, the base of the ladder is 5 m from the wall. Find the angle between the wall and the ladder?
  4. A ladder of length 25 m is resting against a wall, the ladder makes an angle 37 to the wall. Find the distance between the wall and the base of the ladder?
  5. In the following triangle find the angle A B ^ C
  6. In the following triangle find the length of side C D
  7. A ( 5 ; 0 ) and B ( 11 ; 4 ) . Find the angle between the line through A and B and the x-axis.
  8. C ( 0 ; - 13 ) and D ( - 12 ; 14 ) . Find the angle between the line through C and D and the y-axis.
  9. A 5 m ladder is placed 2 m from the wall. What is the angle the ladder makes with the wall?
  10. Given the points: E(5;0), F(6;2) and G(8;-2), find angle F E ^ G .
  11. An isosceles triangle has sides 9 cm , 9 cm and 2 cm . Find the size of the smallest angle of the triangle.
  12. A right-angled triangle has hypotenuse 13 mm . Find the length of the other two sides if one of the angles of the triangle is 50 .
  13. One of the angles of a rhombus ( rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter 20 cm is 30 .
    1. Find the sides of the rhombus.
    2. Find the length of both diagonals.
  14. Captain Hook was sailing towards a lighthouse with a height of 10 m .
    1. If the top of the lighthouse is 30 m away, what is the angle of elevation of the boat to the nearest integer?
    2. If the boat moves another 7 m towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer?
  15. (Tricky) A triangle with angles 40 , 40 and 100 has a perimeter of 20 cm . Find the length of each side of the triangle.

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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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