Download 1 Right Triangles

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MA 1165 - Lecture 15
02/23/09
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Right Triangles
A lot of what we do is based on the properties of a right triangle. In particular, most of the rest of the
semester will be devoted to the trigonometric functions, which will be defined in terms of right triangles.
B
c
A
b
a
C
Figure 1: A right triangle. The angle C is a right angle.
Figure 1 shows an example of a right triangle, which is a right triangle because one of its angles is a right
angle. Remember that right angles measure 90◦, or in radians, π2 radians. We’ll work in degrees for now.
Note that in Figure 1, I’ve labeled the vertices with upper-case letters A, B, and C. The sides are labeled
with lower-case letters corresponding to the opposite vertex. For example, the side a is opposite the vertex
A. I will also use the letters A, B, and C to represent the measures of the angles at these vertices. So in
this case, the angle at vertex C measures 90◦, so I might just say C = 90◦. Similarly, I will use the letters
a, b, and c to represent the lengths of these sides.
Basic Fact 1. The angles of any triangle add up to 180◦. Since one of the angles of a right triangle measures
90◦ , the other two angles must add up to 90◦ .
In Figure 1, angle C is the right angle, so C = 90◦ . Therefore, A + B = 90◦ . For example, if A = 40◦ , then
B must measure 50◦ .
The side opposite the right angle is called the hypotenuse. In Figure 1, c is the hypotenuse. Since the right
angle is always the biggest angle of a right triangle, the hypotenuse is always the longest side. In fact, we
have the following fundamental fact.
Basic Fact 2. The square of the hypotenuse is equal to the sum of the squares of the other two sides. In
Figure 1, we have
c2 = a 2 + b 2 .
(1)
This is called the Pythagorean Theorem.
Example. Suppose in Figure 1, c = 5 and a = 4. The Pythagorean theorem allows us to find the length of
the other side. Plugging into equation above, we get
(5)2 = (4)2 + b2 .
(2)
25 = 16 + b2 ,
(3)
9 = b2 ,
(4)
±3 = b.
(5)
We can solve for b by simplifying,
moving the 16 to the left side,
and taking the square root of both sides,
For now, we’re only interested in the length of the side b, so we’ll just take the positive solution b = 3.
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2 QUIZ 15A
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Quiz 15A
1.
In Figure 1, suppose A = 55◦. What is B?
2.
In Figure 1, suppose a = 3 and c = 7. Find b.
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The Trigonometric Functions
As we’ve seen, once we know the measure of one angle in addition to the right angle, then we can figure out
the measure of the third angle. All right triangles with an angle measuring 30◦, for example, have another
angle measuring 60◦ , and they all have the same shape. We say that they’re all similar.
Basic Fact 3. The ratios between corresponding sides of similar triangles are the same.
The ratios in right triangles are used to define the trigonometric functions. In Figure 1, the sides a and c are
adjacent to the angle A. Since the side c already has a name, the hypotenuse, if we say the side adjacent to
A, we mean b. Similarly, a is adjacent to B. We’ve already said that a is opposite A, and b is opposite B.
It’s common to use the Greek letter θ (“theta”) to represent an angle. So suppose θ is one of the smaller
angles of a right triangle. Then we will define the following.
opposite
hypotenuse
tan(θ) = opposite
adjacent
sec(θ) = hypotenuse
adjacent
adjacent
cos(θ) =
hypotenuse
adjacent
cot(θ) = opposite
hypotenuse
csc(θ) = opposite
sin(θ) =
(6)
The names for the trigonometric functions are
sin ≡ sine
tan ≡ tangent
sec ≡ secant
cos ≡ cosine
cot ≡ cotangent
csc ≡ cosecant
(7)
Example. Suppose in Figure 1, we have that a = 3 and b = 5. By the Pythagorean theorem, we have that
c2 = 32 + 52 = 9 + 25 = 34.
It follows that
c=
(8)
√
34
(9)
(we won’t worry about the negative square root here). For the angle A, a is opposite, b is adjacent, and c is
the hypotenuse. Therefore,
sin(A) =
opposite
a
3
= = √ ≈ 0.514495755.
hypotenuse
c
34
(10)
We can compute the values of all the trig functions at A just as easily.
sin(A) =
√3
34
cos(A) =
√5
34
tan(A) =
3
5
cot(A) =
5
3
sec(A) =
csc(A) =
√
34
5
√
(11)
34
3
In decimal form, rounded to four decimal places these are
sin(A) = 0.5145
tan(A) = 0.6000 sec(A) = 1.1662
(12)
cos(A) = 0.8575 cot(A) = 1.6667
csc(A) = 1.9437
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4 QUIZ 15B
Since the hypotenuse is the longest side, the sine and cosine must always be less than 1. Similarly, the secant
and cosecant must be larger than 1. The tangent and cotangent can range anywhere between 0 and ∞.
Example. An extreme case is a triangle where one of the sides adjacent to the right angle is small and the
other is big. For example, suppose that a = 100 and b = 1. This would be a really tall skinny triangle. From
the Pythagorean theorem,
c2 = 1002 + 12 = 10000 + 1 = 10001,
(13)
√
so c = 10001. The values of the trig functions at A are
sin(A) =
√ 100
10001
cos(A) =
√ 1
10001
tan(A) =
100
1
cot(A) =
1
100
sec(A) =
√
10001
1
csc(A) =
√
10001
100
(14)
My calculator gives me the following.
sin(A) = 0.999950003
tan(A) = 100
sec(A) = 100.0049999
(15)
cos(A) = 0.009999500 cot(A) = 0.01 csc(A) = 1.000049999
Here the angle A is approximately 89.43◦, which is about as big as it could get in a right triangle. The angle
B, therefore, is about 0.57◦, which is really small. At B, the trig functions are as follows.
sin(B) = 0.009999500
tan(B) = 0.01 sec(B) = 1.000049999
(16)
cos(B) = 0.999950003 cot(B) = 100
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Quiz 15B
For problems 1-5, suppose that in Figure 1, a = 9 and b = 7.
1.
Find c.
2.
Find sin(A).
3.
Find sec(A).
4.
Find tan(B).
csc(B) = 100.0049999
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5 HOMEWORK 15
5.
Find tan(A).
For problems 5-8, suppose that in Figure 1, b = 2 and c = 12.
6.
Find a.
7.
Find csc(A). (Give answer in approximate decimal form.)
8.
Find cos(B).
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Homework 15
All of these problems refer to a right triangle like the one in Figure 1 with C = 90◦.
1.
If B = 27◦, what is A?
(a) 70◦
2.
(b) 20◦
(c) 63◦
(d) Impossible for a right triangle.
(e) none of these
(d) Impossible for a right triangle.
(e) none of these
If A = 110◦, what is B?
(a) 70◦
(b) 20◦
(c) 63◦
For problems 3-6, suppose that a = 2 and c = 3.
Find b.
√
√
(b) 5
(a) 13
3.
4.
(a)
5.
(a)
6.
(a)
(c)
√
221
(d)
√
21
(e) none of these
Find sin(A).
√
13
3
(b)
√
5
2
(c)
√
5
3
(d)
2
3
(e) none of these
(d)
2
3
(e) none of these
(d)
2
3
(e) none of these
Find cos(A).
√
13
3
(b)
√
5
2
(c)
√
5
3
Find tan(B).
√
13
3
(b)
√
5
2
(c)
√
5
3
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5 HOMEWORK 15
For problems 7-10, suppose that a = 11 and b = 10.
Find c.
√
√
(a) 13
(b) 5
7.
8.
(a)
9.
(a)
10.
(a)
√
221
(d)
√
21
(c)
√10
221
(d)
√
221
10
(e) none of these
(c)
√10
221
(d)
√
221
10
(e) none of these
(c)
√10
221
(d)
√
221
10
(e) none of these
(c)
(e) none of these
Find sin(B).
√10
21
(b)
√
21
10
Find sec(A).
√10
21
(b)
√
21
10
Find csc(B).
√10
21
(b)
√
21
10