Download Trigonometric Ratios of Any Angle

Trigonometric Ratios of Any Angle
There are some angles that occur sufficiently frequently that their trigonometric
ratios should be known exactly, rather than as decimal approximations. Angles that
measure 300, 450, and 600 are called special angles. Right triangles in which they
are found are then called special triangles.
INVESTIGATE:
∆ABC is an equilateral triangle with side lengths of 2 units. AD
bisects  BAC to form two congruent triangles, ∆ABD and
∆ACD. ∆ABD and ∆ACD are called 300- 600-900 triangles.
1. What is the length of BD?
2. Determine the length of AD. Express your answer in radical
form.
3. Repeat questions 1 and 2, beginning with equilateral triangles with the
following side lengths. Express the length of AD as a mixed radical and
complete the chart below.
Equilateral Triangle
2 units
BD
AD
4 units
10 units
4. Describe the relationship between the side lengths in any 300- 600-900 triangle.
5. Use one of the 300- 600-900 triangles to determine the exact values of the
sine, cosine and tangent of 300 and 600.
Angle
Exact
300
600
Sine
Cosine
Calculator Exact
Calculator
Exact
Tangent
Calculator
∆DEF is an isosceles right triangle whose equal sides, DE and
DF, have side lengths of 1 unit. ∆DEF is called a 450- 450-900
triangle.
6. Determine the length of EF. Express the answer in radical
form.
7. Repeat question 6, beginning with right isosceles triangles
whose equal sides have the following lengths. Express the
length EF as a mixed radical.
Right Isosceles Triangle
1 units
EF
2 units
6 units
8. Describe the relationship between the lengths of the sides of any 450- 450900 triangle.
9. Use one of the 450- 450-900 triangles to determine the exact values of the
sine, cosine and tangent of 450.
Angle
Exact
450
Sine
Cosine
Calculator Exact
Calculator
Exact
Tangent
Calculator
Rotation Angles
Our work in trigonometry has so far been limited to angles between 00 and 900. If
an angles has its vertex at the origin in the XY plane and its initial arm along the
positive X-axis, then the measure of an angle can be extended
beyond 900. We do this by rotating the terminal arm of the
angle about the origin.
The XY-plane is divided into four quadrants, numbered as shown:
The
angle of rotation is the angle created when a line
segment is rotated about its vertex to a terminal position.
An angle is in standard position if its vertex is at the origin
in the XY plane, its initial arm is on the positive X-axis and its
terminal arm is a rotation of the initial arm about the origin.
If the rotation is counterclockwise, the angle has positive measure. If the
rotation is clockwise, the angle has negative measure. The rotation is usually
indicated by a directed arrow starting from the positive X-axis.
For a rotation angle, the acute angle built to the X-axis is called a reference
angle. For each angle given below, state the reference angle and the value of x.
Two angles in standard position are coterminal if they have the same terminal arm.
Coterminal angles differ by a multiple of 3600. For example, 800 is coterminal with
4400, since 800 + 3600 = 4400.