Download Aim: How do we apply the sine and cosine ratio for

Aim: How do we apply the inverse trig?
Objectives: to apply the inverse trig ratios to calculate unknown angles when given side lengths in a right
triangle
Do Now: In the accompanying diagram, x represents the length of a ladder that is leaning against a wall of a
building, and y represents the distance from the foot of the ladder to the base of the wall. The ladder makes a
60° angle with the ground and reaches a point on the wall 17 feet above the ground. Find the number of feet in x
and y to the nearest hundredth place with and without trig ratios.
tan 60 
y
17
y
17
 9.81
tan 60
17
x
x  19.63
sin 60 
Since it is a 30-60-90 triangle, we can easily
17
34
determine that y 
,x 
3
3
Lesson Development: Previously, we used trig ratios to find the missing sides given at least one side and an
angle. Now when given at least two sides, we can find the any missing angle in the right triangle. To
accomplish that, we need to introduce the inverse trig functions.
EX: Find the measure of each missing angle to the nearest degree.
Sin A 
15
17
A  62
Therefore mB  28 .
Show that the angles measures remain the same by using
the inverse tangent or cosine function.
Make sure the calculator MODE is set to degrees.
EX1: Find the value of x to the nearest degree.
420
2000
420
x  tan 1 (
)  12
2000
tan x 
cos x 
3
7
3
x  cos 1 ( )  65
7
EX2: Verify that the two missing angles are 45 .
30
30
1
C  tan (1)  45
tan C 
30
30
1
A  tan (1)  45
tan A 
EX3: The grade of a road is the ratio of its rise to its run and is usually given as a decimal or percent. Find the
angle to the nearest degree that the road makes with the horizontal if the grade is 4%.
Let the angle be represented by x, then tan x  0.04 , which gives the angle to be 2
EX4: A rhombus has diagonals of length 4 and 10. The diagonals of a rhombus are perpendicular. So we
Find the angles of the rhombus to the nearest can use
degree.
2
tan(BCA) 
5
1
BCA  tan (0.4)  22
C  A  44
2
5
5
2
1
BCA  tan (2.5)  68
tan(CBA) 
B  B  136
HW#9: P309: 19 – 20, 23
P314 – 316: 7, 8, 13, 25
Solutions
P309:
19) x  4
P314-316:
7) v  26
8) v  67
13) a) 115
25)
20) grade = 14%
b) y  40
x = 10.7
23) x = 87; diagonal = 174 cm
c) yes; 115  10.7
Statements
1) STR is a right triangle
2) ( ST )2  (TR)2  ( RS )2
ST
TR
3) sin R 
; cos R 
;
RS
RS
( ST )2
(TR )2
2
4) (sin R)2 
;
;
(cos
R
)

( RS )2
( RS )2
Reasons
1) given
2) Pythagorean theorem
( ST )2 (TR )2 ( ST ) 2  (TR) 2 ( RS ) 2



1
( RS ) 2
( RS ) 2
5) ( RS )2 ( RS )2
5) algebra and substitution
(sin R) 2  (cos R) 2  1
3) Def of trig ratios
4) squaring both sides
Aim: How do we apply the inverse trig functions?
Do Now: In the accompanying diagram, x represents the length of a ladder that is leaning against a wall of a
building, and y represents the distance from the foot of the ladder to the base of the wall. The ladder makes a
60° angle with the ground and reaches a point on the wall 17 feet above the ground. Find the number of feet in x
and y to the nearest hundredth place with and without trig ratios.
Lesson Development: Previously, we used trig ratios to find the missing sides given at least one side and an
angle. Now when given at least two sides, we can find the any missing angle in the right triangle. To
accomplish that, we need to introduce the inverse trig functions.
EX: Find the measure of each missing angle to the nearest degree.
Make sure the calculator MODE is set to degrees.
EX1: Find the value of x to the nearest degree.
EX2: Verify that the two missing angles are 45 .
EX3: The grade of a road is the ratio of its rise to its run and is usually given as a decimal or percent. Find the
angle to the nearest degree that the road makes with the horizontal if the grade is 4%.
EX4: A rhombus has diagonals of length 4 and 10.
Find the angles of the rhombus to the nearest degree.
2
5