Final Culminating: Trigonometry

Final culminating: Trigonometry

[pic 1]

                                                                                Sefera Yared

                                                                                  Course:

                                                                              Mr. Kanhai:

What’s trigonometry?

• Trigonometric is a branch of mathematics that studies triangles and the relationship between their sides and the angles between these sides.
• The words “Trigonometry” is derived from the Greek words “Tri” (meaning three), “Gon” (meaning sides) , “Metron” (meaning measure)

                                   [pic 2]

4.1: Special angles

* Unit circle: a circle with centre at the origin and a radius of 1 unit.

* Initial arm: first arm, or ray, of an angle drawn on a Cartesian plane that meets the other (terminal) arm of the angle at origin.

* Terminal arm: the arm of an angle that meets the origin and rotates around the origin counter clock wise to form a positive angle or clockwise to form a negative angle

* Angle in standard position: the position of an angle when its initial arm is on the positive x-axis and its vertex is at the origin.

* Reference angle: the acute angle between the terminal arm and the x-axis of an angle in standard position.

Example: 1 [pic 3]

Trigonometric ration for 0, 90,180, and 270

Use a unit circle to find exact values of the trigonometric ration for 0, 90,180, and 270.

Solution:[pic 4]

Sin 0=0, cos 0=1, tan 0= 0

Sin 90=1, cos 90=0, tan 90= undefined

Sin 180=0, cos 180= -1, tan 180=0

Sin 270=-1, cos 270=0, tan=undefined

Example: 2[pic 5]

An air traffic controller observes that a ValuAir flight is 20 km due east of the controller tower, while a first class air flight is 25 km in a direction 10 degree west of north from the control tower.

1. What is the angle of separation of the two air craft as seen from the tower?

Solution: from east to north is an angle of 90 degree. A further angle of 10 degree results in an angle of separation of 100 degree,[pic 6]

Practice: 1[pic 7]

In a table, summarize the exact trigonometric ratio for the angles 0, 30, 45, 60

Practice: 2

1.  When using a unit circle to find trigonometric ratios for 135, a reference angle of 45 is used. What reference angle should you use to find the trigonometric ratios for 120?

Solution: we use 60 degree.

4.2: Co-terminal and Related angles

Conterminal Angles are angles who share the same initial side and terminal sides.

Example 1:

1.  Given that sin A= 3/5 and that
2. Determine the primary trigonometric ratios for another angle between 0 and 360 that has the same sine value.

Solution:

1. Since sin A= 3/5, possible values of y and r are 3 and 5. Therefore let y=3 and r=5

x2 + y2=r2

x2+3 2=5 2 

       X 2=16

       x= ±4  

Since < A lies in the first quadrant, x=A

Cos A= x/r                                   tan A=y/x

          = 4/5                                            = ¾

1. The sine ratio is positive in the first and second quadrants. The point that defines an angle with the same sine is (-4, 3). Let

Sin B = y/t           Cos B= x/t              Tan B= y/x

       = 3/5                   = -4/5                          =3/-4

                                   = -4/5                          =-3/4

Example: 2

1. Determine another angle between 0 and 360 that has the same tangent as < A in example 1. What is the relationship between this angle and < A?

Solution

Determine another angle, b, such that tan b= ¾. Since the tangent ratio is positive in the third quadrant, the coordinates of the required point are (-4,-3).

          B= 180+

            =180+37

            =217

Practice: 1

Determine the exact primary trigonometric ratios for each angle. You may wish to use a unit circle to help you.

c)

Solution: 2

a) Sin A= 1/ √2, cos A= 1/ √2, tan A= -1[pic 8]

b) Sin B=- √3/2, cos B=-1/2, tan B= √3

c) Sin C= 0, cos C=-1, tan C=0

d) Sin D= 1/ √2, cos D= 1/ √2, tan D= 1

Practice: 2

Determine another angle that has the same trigonometric ratio as each given angle.

a) Cos 45                            b) sin 150                     c) tan 300                        d) sin 100                    e) cos 230                         f) tan 350