Introduction to right triangle trigonometry

1

• Introduction to right triangle trigonometry
• Define three basic trigonometric functions sine,
  cosine, and tangent
• Using right triangle trigonometry to solve
  problems

2
Trigonometry

• Trigonometry is the study of the relationships
  between the sides and the angles of triangles
• In any right triangle with the same angle
  measure, the ratio of the lengths of the legs is
  the same
• This follows from AA similarity,
• since all such triangles are similar
• The legs are identified based on where
• they are in relation to the angle
• One leg is the side opposite the angle
• The other leg is the side adjacent to the angle

3
Trigonometry

• The tangent of an angle in a right triangle is
  the ratio of the length of the opposite side to
  the length of the adjacent side
• If the length of one leg and the measure of one
  angle of a right triangle is known, the tangent
  provides a method of finding the length of the
  other leg
• Given the triangle at right, the tangent of 31
  is 3/5
• Use the tangent to find the height of the tree in
  this picture

4
Trigonometry

• In addition to the tangent, there are two other
  trigonometric functions defined in this chapter
• The sine of an angle in a right triangle is the
  ratio of the length of the opposite side to the
  length of the hypotenuse
• The cosine of an angle in a right triangle is the
  ratio of the length of the adjacent side to the
  length of the hypotenuse
• Trig functions are usually abbreviated tan, sin,
  and cos
• Scientific calculators usually have buttons for
  these three trigonometric functions
• Before calculators, the values were calculated by
  hand, and books were published containing tables
  giving the values of trig functions for different
  angles

5
Trigonometry

• The mnemonic SOH-CAH-TOA is sometimes used to
  remember the relationships between the sides
  of a triangle and trig functions
• Consider angle A in the triangle at right
• This is a function, not multiplication!
• This is read as sine of A, cosine of A,
  tangent of A
• Note that the leg opposite one angle is the leg
  adjacent to the other angle

6
Trigonometry

• Trigonometry provides another way to use indirect
  measurement with right triangles
• Given a distance and an angle in a right
  triangle, the length of any other side can be
  found using sine, cosine, or tangent functions
• Surveyors use trigonometry to calculate distances
  in construction
• Example
• Find the length of the hypotenuse of a right
  triangle if an acute angle measures 20 and the
  side opposite the angle measures 410 feet
• First, make a sketch of the problem
• Determine which trig function to use
• SOH-CAH-TOA Sine Opposite/Hypotenuse
• Set up an equation and solve for x

7
Trigonometry

• If two sides of a right triangle are known,
  inverse trig functions can be used to find the
  angle measures
• Scientific calculators can also find inverse trig
  functions
• The inverse functions are written as sin-1x,
  cos-1x, and tan-1x
• If sinA x, then A sin-1x
• For example, to find the angle with a sine of ½
• sinA ½ A sin-1(½) A 30
• Example
• Use an inverse trig function to find the angle
  opposite the shorter leg of a right triangle with
  legs of 8 and 15 inches
• First, make a sketch of the problem
• Use inverse tangent to find mÐA
• tanA 8/15 A tan-1(8/15) A 28

8
Problem Solving with Right Triangles

• Two terms used when making observations using
  angles
• Angle of elevation is the angle looking up from
  horizontal
• Angle of depression is the angle looking down
  below horizontal

• The angle of elevation for the person at left is
  equal to the angle of depression for the person
  at right
• Example
• If the angle of elevation from the sailboat to
  the top of the lighthouse is 16, and the
  lighthouse is 121 feet tall, how far is the boat
  from shore?