Download Using Trigonometric Ratios

USING
TRIGONOMETRIC
RATIOS
Module 11, Lesson 5
Trigonometric Identities
Reference Angles
• When an angle is
graphed on the
coordinate plane,
the positive,
acute angle
formed by the
terminal side and
the x-axis is
called a
reference angle.
• They form a
bowtie!
Graph Angles in the Coordinate Plane
Plot an angle in the
coordinate plane
1. Sketch the angle
with its initial side
on the positive xaxis
2. Construct a right
triangle from the
terminal side to the
x-axis, making the
reference angle
inside the triangle.
Trigonometry Ratios in the Coordinate
Plane
When you sketch a right triangle in the coordinate plane,
the trig ratios are determined using the sides in relation to the
reference angle. The legs are x and y values, and the
hypotenuse is r (radius of the circle).
Signs of Trigonometry Ratios in the
Coordinate Plane
• Trigonometric ratios can be written in terms of x, y, and r.
• x and y can be positive, negative, or 0.
• Trigonometric ratios can be positive, negative, 0, or
undefined. Recall the unit circle for the quadrant angles.
Find a trig ratio given a trig ratio
1. Make a sketch of a right triangle in the correct quadrant.
Be sure that you connect to the x axis and make the
reference angle at the origin.
2. Label the sides according to the trig ratio given. Be sure
to put any + or - signs on the correct side. [SOHCAHTOA]
3. Find the third side (hypotenuse) using the Pythagorean
Theorem. [a² + b² = c²]
4. Now use the trig ratios to find the exact value.
[SOHCAHTOA]
EXAMPLE
1
• Find cos  and tan  given sin   in quadrant I.
5
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
• Since sin  = opposite/hypotenuse, you label those 2
sides.
1 b  5
4. Calculate the adjacent side using a² + b² = c² 1  b  25
2
2
2
b 2  24
a 2 5
cos   
h
5
0
1
5
5
tan   


a 2 5 5 10
b2 5
2
EXAMPLE
1
• Find sin  and tan  given cos    in quadrant II.
4
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
• Since cos  = adjacent/hypotenuse, you label those 2
sides.
(1)  b  4
4. Calculate the opposite side using a² + b² = c² 1  b  16
2
2
b 2  15
o
15
sin   
h
4
0
15
tan   
  15
a
1
b  15
2
2
EXAMPLE
3
• Find sin  and cos  given tan  
in quadrant III.
4
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
• Since tan  = opposite/adjacent, you label those 2 sides.
4. Calculate the hypotenuse using a² + b² = c² 3  4  c
2
2
9  16  c 2
o
15
sin   
h
4
0
15
tan   
  15
a
1
25  c 2
c5
2
EXAMPLE
1
• Find cos  and tan  given sin    in quadrant 4.
5
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
• Since sin  = opposite/hypotenuse, you label those 2
sides.
(1)  b  5
4. Calculate the adjacent side using a² + b² = c² 1  b  25
2
2
a 2 5
cos   
h
5
0
1
5
5
tan   


a 2 5 5
10
b 2  24
b2 5
2
2