Download Trigonometric Functions of Acute Right Triangles Lesson Plan

Trigonometric Functions of Acute Right Triangles
Lesson Plan
By: Douglas A. Ruby
Class: Pre-Calculus II
Date: 10/10/2002
Grades: 11/12
INSTRUCTIONAL OBJECTIVES:
At the end of this lesson, the student will be able to:
1. Derive the values of the 6 trigonometric functions given an acute right triangle described
using a standardized terminology.
2. Recall and use all 6 trigonometric function values for the 30°/60°/90° right triangle and the
45°/45°/90° right triangle.
3. Complete the 3 angles and 3 sides of a right triangle, given an incomplete description, using
standard notation and the appropriate trigonometric functions.
4. Demonstrate correct application of basic trig functions when using either simple angles (pen
and pencil) or complex angles (using a calculator).
Relevant Massachusetts Curriculum Framework
PC.P.3 - Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent,
cosecant, secant, and cotangent). Relate the functions to their geometric definitions.
MENTAL MATH – (5 Minutes)
Question 1: How many degrees is the angle A below? 30o Angle B? 60o
B
2
A
1
3
C
Question 2: How many degrees is the angle B below? 45o What is the length of side c? 3*sqrt(2)
B
c
A
3
3
C
Page 1
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
CLASS ACTIVITIES – (Note: 45 Minute Lesson Plan)
1. Trigonometric Ratios
The study of trigonometric ratios (or functions) is based on the relationships of the angles and
sides of a right triangle. We will use the “standard” triangle as shown below. It is customary to
call the right angle (the 90o degree angle) angle C. The acute angles are labeled A and B.
B
c
A
a
b
a = opposite side
b = adjacent side
c = hypotenuse
C
We remember that acute angles are those angles that are less than 90o. Given the orientation of
this triangle, we define side |a to be the side opposite angle A, side |b is the side adjacent to
A, and side |c is the hypotenuse.
Based on these definitions, the sine of angle A (written as sin A) is the length of side opposite
A divided by the length of the hypotenuse. This is written as:
sin A =
side _ opposite _ A
hypotenuse
a
c
In fact, there are six trigonometric ratios. These include sine, cosine, tangent, cosecant, secant,
and cotangent. The formulas for these six functions with respect to angle A are:
sin A =
side _ opposite _ A
hypotenuse
a
c
csc A =
hypotenuse
side _ opposite _ A
c
a
cos A =
side _ adjacent _ A
hypotenuse
b
c
sec A =
hypotenuse
side _ adjacent _ A
c
b
tan A =
side _ opposite _ A
side _ adjacent _ A
a
b
cot A =
side _ adjacent _ A
side _ opposite _ A
b
a
a
is only with respect to
c
B, we would need to look at the ratio of:
It is important to realize that the sin A =
the sine of angle
sin B =
side _ opposite _ B
hypotenuse
b
b
. Notice that sin B =
= cos A.
c
c
Page 2
A. If we wanted to know
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
We will develop a mnemonic for memorizing the six trigonometric functions later, but based on
this overview, let’s look at a triangle and see if we can identify the six trigonometric functions.
5
3
4
a) For this triangle, what are the six trigonometric functions for ?
sin = 3/5
cos = 4/5
tan = 3/4
csc = 5/3
sec = 5/4
cot = 4/3
b) What are the six trigonometric functions for ? (Be careful to reorient yourself!)
sin = 4/5
cos = 3/5
tan = 4/3
csc = 5/4
sec = 5/3
cot = 3/4
Notice that the cosecant, secant, and cotangent are the reciprocals of the sine, cosine, and
tangent, respectively. This means that:
csc
1
,
sin
sec
1
,
cos
cot
1
,
tan
2. SOHCAHTOA
A helpful mnemonic for memorizing the basic trigonometric functions is SOHCAHTOA. This
means that:
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent.
Once we know the sine, cosine, and tangent functions, the cosecant, secant, and cotangent are
just the reciprocal functions as defined earlier. Lets try an example:
Page 3
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
7
, draw the triangle represented by the angle and the Opposite and
4
Adjacent sides of the triangle. Use the Pythagorean Theorem to .solve the hypotenuse, then use
SOHCAHTOA to provide the other 5 trigonometric functions.
Try Now: If tan
Hyp.
Opp.=7
Hyp2 = Adj.2+Opp2 = 72 + 42 = 49 + 16 = 65 = (13 x 5)
Hyp2 = 65
Hyp = 65
Adj.=4
Sine = Opposite/Hypotenuse = 7 /
65
Cosine = Adjacent/Hypotenuse = 4 / 65
Tangent = Opposite/Adjacent.= 7 / 4
Cosecant = reciprocal of sine = 65 / 7
Secant = reciprocal of cosine = 65 / 4
Tangent = reciprocal of tangent = 4 / 7
3. Standard 30o/60o/90o and 45o/45o/90o triangles
Knowing the trigonometric functions of standard 30o/60o/90o and 45o/45o/90o triangles is
important. From our previous section (and review from our Mental Math), lets recall the basic
form of the 30o/60o/90o right triangle. This looks like:
B=60
2
A=30
In addition:
1
3
Therefore:
sin 30 = ½
cos 30 = 3 / 2
csc 30 = 2
sec 30 = 2 / 3
tan 30 = 1 / 3
cot 30 =
sin 60 = 3 / 2
cos 60 = ½
tan 60 = 3
csc 30 = 2 / 3
sec 60 = 2
cot 60 = 1 / 3
3
C=90
Page 4
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
For the 45o/45o/90o triangle, we have:
B
1
2
A
Therefore:
sin 45 = 1 / 2
cos 45 = 1 / 2
tan 45 = 1
1
csc 45 = 2
sec 45 = 2
cot 45 = 1
C
Regardless of the length of the sides of the 30/60/90 or 45/45/90 triangle, the trigonometric ratios
for the six functions will remain the same. Let’s use our calculators to try some examples.
4. Using your Calculator
Let’s now pull out our calculators and try a few examples. First, who has a graphing calculator
(such as a TI-83+)? Who has a non-graphing scientific calculator? (ex: TI-3x) Let’s try some
examples. I’d like you to first make sure that your calculator is set up for entering angles in
degrees instead of radians (more later…)
First, use your calculators to approximate the following ratios:
2
1.414
1
2
.707
3
1.732
1
3
.577
3/2
.866
2/ 3
1.154
Now, let’s use our calculators to look at the following:
sin 45o = .707
cos 30o = .866
tan 30o = .577
csc 45o = 1.414
sec 30o = 1.154
cot 30o = 1.732
Lets try a few more:
sin 10o = .1736
cos 57.295o = .5403
sin o = .0548
csc 14o = 4.134
sec 57.295o = 1.8508
cot o = 18.2195
5. Properties of Trigonometric Functions
Further, the two opposite acute angles of a triangle are complementary to each other (i.e A=90B). We observed earlier that sin B = cos A. The sine and cosine functions are complementary
to each other since sin B = cos (90-B). The same relationship exists between tangent and
cotangent, and secant and cosecant respectively.
Page 5
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
This can be mathematically stated as follows:
sin
cos(90
),
tan
cot(90
Example: So, if you are given sin
=
),
sec
csc(90
)
1
, what are the values for the six trigonometric functions
2
for , and 90- ?
Answer:
sin
=
1
, cos
2
sin 90- =
=
3
, tan
2
=
1
3
, csc
3
1
, cos 90- = , tan 90- =
2
2
= 2, sec
=
3 , csc 90- =
Now we will review our homework.
Page 6
2
3
2
, cot
3
=
3
, sec 90- = 2, cot 90- =
1
3
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
HOMEWORK:
Solve the following four problems using your calculator
1. cos 18°
.95106
2. sin 37°
.601815
3. tan 2.6°
.04541
4. cos 34.8°
.82115
Given the following two triangles, what are the six trigonometric functions for the given angle?
5.
sin = 8/17
cos = 15/17
tan = 8/15
csc = 17/8
sec = 17/15
cot = 15/8
6.
sin = 15/17
cos = 8/17
tan = 15/8
csc = 17/15
sec = 17/8
cot = 8/15
Using the information given, solve the following three triangles (i.e. provide sides a, b, c, and
angles A, B, and C.
7. Solve the right triangle below:
A = 30o
B = 60o
C =90o
a=3
b = 3 * sqrt(3)
c=6
Page 7
Trigonometric Functions of Acute Right Triangles – Mr. Ruby
8. Solve the right triangle below:
A =45o
B =45o
C =90o
a = 5 * sqrt(2)
b = 5 * sqrt(2)
c = 10
9. Solve the triangle below:
A = 90o-26.7o=63.3o
B =26.7o
C =90o
a = 0.17 / tan 26.7o .338
b = 0.17
c = 0.17 / sin 26.7o .378
Note: using Pythagorean Theorem, c = sqrt ( .3382+.172)
10. What is the length of the ladder “h” in the diagram below:
Using the Pythagorean Theorem:
h2 = 72 + 42
h = 65 8.06225 ft.
Also,
tan = 7 / 4
= tan –1(7/4) 60.26o
sin 60.26o = 7 / h
h = 7ft / sin 60.26o 8.0618 ft.
Page 8
.378
This document was created with Win2PDF available at http://www.daneprairie.com.
The unregistered version of Win2PDF is for evaluation or non-commercial use only.