Download Trigonometric Ratios

Trigonometric Ratios
Finding more relationships
Pages 25, 26, 27 and 28
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Right Triangles
2n
n
n√2
n√3
n
n
Understanding Trigonometric Ratios
page 25 1. Notice the patterns...
sin 30° = 1
2
sin 60° = √3
2
sin 45° = 1
√2
cos 30° = √3
2
cos 60° = 1
2
cos 45° = 1
√2
tan 30° = 1
√3
tan 60° = √3
1
tan 45° = 1 =1
1
Understanding Trigonometric Ratios
page 25 2.
0.6691 is the sine of 42 degrees, which
means it is the ratio of the leg opposite
the 42 degree angle in a right triangle to
the hypotenuse in that right triangle
(sine = opposite/ hypotenuse).
Understanding Trigonometric Ratios
page 25 3. #1
Triangles CBD and EGF are similar right
triangles (by angle- angle similarity
postulate) and similar triangles have
proportional corresponding sides.
Thus, the ratio of CD/CB has to be the
same as EF/EG because of the
similarity relationship between the
triangles.
Understanding Trigonometric Ratios
page 25 3.#2
CB/CD = 0.9135
Since CB/CD is the leg adjacent to the
24 degree reference angle divided by
the length of the hypotenuse of the
right triangle and this ratio is called the
cosine of the reference angle, Thomas
must have typed cos(24°) = 0.9135
Understanding Trigonometric Ratios
page 26 4
Sine ratio for an angle is opposite side
divided by hypotenuse, and cosine ratio is
adjacent side divided by hypotenuse and
so the hypotenuse is the denominator in
each ratio. The hypotenuse is always the
longest side of a right triangle, so for sine
and cosine the ratio will be a value
between 0 and 1.
Understanding Trigonometric Ratios
page 26 5
How is tangent the same as slope?
Tangent is
Opposite leg vertical distance
rise
=
=
Adjacent leg horizontal distance
run
Hence tangent and slope are the same.
Understanding Trigonometric Ratios
page 26 6.
At 45°, the tangent value =1.
a) What does that mean about
the opposite and adjacent
sides?
Opposite and adjacent sides are
the same length (remember how
we cut up a square to make 2
45-45-90 triangles).
Understanding Trigonometric Ratios
page 26 6.
For angles greater than 45°, the
tangent value >1.
b) What does that mean about
the opposite and adjacent sides?
Opposite side is always bigger
than the adjacent side (greater
rise than run).
Opp
Adj
Understanding Trigonometric Ratios
page 26 6. (another one)
Jesse claims that AB is the
opposite side. Is he correct?
Explain.
It depends on which reference A
angle we are interested in. AB is
opposite reference angle o, but
adjacent to reference angle θ. CB
is opposite reference angle θ.
C
o
θ
B
Understanding Trigonometric Ratios
page 26 7
What does similarity have to do with A θ
trigonometry? Everything.
Trigonometry is all about the fixed relationships
between the side lengths and the angles in right
triangles and the constant ratios for particular
reference angles only exist because the triangles
are similar by Angle-angle similarity.
C
o
B
Understanding Trigonometric Ratios
page 27 1. Notice the patterns...
sin 10° = cos 80°
sin 9° = cos 81°
sin 8° = cos 82°
sin 5° = cos 85°
Etc. etc.
Use your full tables to notice other
pairs that have the same values.
1. What do the angles that have the same value
for cosine and sine have in common?
When sine and cosine values are the
same the reference angles are
complementary (add up to 90 degrees).
2. Why does
sin θ° = cos (90-θ)°
θ and (90-θ) are complements. (They are
the same side of a right triangle with
reference angle θ.
Sinθ° = a
c
a
c
θ°
2. Why does
sin θ° = cos (90-θ)°
θ and (90-θ) are complements. (They are
the same side of a right triangle with
reference angle θ.
cos(90-θ)° = a
c
90 - θ°
a
c
θ°
3. Solve the following:
a)
b)
c)
d)
sin 42° = cos 48°
cos 12° = sin 78°
sin 45° = cos 45°
cos 0° = sin 90°
48 + 42 = 90
12 + 78 = 90
45 + 45 = 90
0 + 90 = 90
90 - θ°
a
c
θ°
3. Solve the following:
a)
b)
c)
d)
e)
f)
sin 42° = cos 48°
cos 12° = sin 78°
sin 45° = cos 45°
cos 0° = sin 90°
cos 65° = sin 25°
sin 78.5° = cos 11.5°
48 + 42 = 90
12 + 78 = 90
45 + 45 = 90
0 + 90 = 90
65 + 25 = 90
78.5 + 11.5 = 90
4. Solve for the unknown:
a) sin (x - 5)° = cos 35°
x - 5 + 35 = 90
x + 30 = 90
x = 60
4. Solve for the unknown:
b) sin (2x - 17)° = cos (x - 4)°
2x - 17 + x - 4 = 90
3x - 21 = 90
3x = 111
x = 37
4. Solve for the unknown:
c) sin (x)° = cos (x )°
x + x = 90
2x = 90
x = 45
4. Solve for the unknown:
d) sin (¾ x)° = cos (¼ x )°
¾ x + ¼ x = 90
x = 90
4. Solve for the unknown:
e) sin (5x - 22)° = cos (x - 10 )°
5x - 22 + x - 10 = 90
6x - 32 = 90
6x = 122
x = 61/ 3 or 20 ⅓
4. Solve for the unknown:
f) sin (¾ x - 3)° = cos 66°
¾ x - 3 + 66 = 90
¾ x + 63 = 90
¾ x = 27
x = 4(27) / 3
x = 36
Geometry Unit 3B - Trigonometric Ratios of Acute
angles - page 28
1. a =5, b =12
c2 = 52 + 122
c2 = 25 + 144
c = √169 = 13
B
5
C
13
12
A
Geometry Unit 3B - Trigonometric Ratios of Acute
angles - page 28
1.
sinA= 5
13
cosA= 12
13
tanA= 5
12
B
sinB= 12
13
cosB= 5
13
tanB= 12
5
5
C
13
12
A
Geometry Unit 3B - Trigonometric Ratios of Acute
angles - page 28
2. a =3, b =14
c2 = 32 + 42
c2 = 9 + 16
c = √25 = 5
B
3
C
5
4
A
Geometry Unit 3B - Trigonometric Ratios of Acute
angles - page 28
2.
sinA= 3
5
cosA= 4
5
tanA= 3
4
B
sinB= 4
5
cosB= 3
5
tanB= 4
3
3
C
5
4
A
Geometry Unit 3B - Trigonometric Ratios of Acute
angles - page 28
a =3, b =8
c2 = 32 + 82
c2 = 9 + 64
c = √73
B
3
C
√73
8
A
Geometry Unit 3B - Trigonometric Ratios of Acute
angles - page 28
B
sinA= 3 = 3√73
√73
73
cosA= 8 = 8√73
√73
73
tanA= 3
8
sinB= 8√73
3
73
cosB= 3√73
C
73
tanB= 8
3
√73
8
A
Use the given trigonometric ratio to
find exact values for sinA, cosA or tan
A (don’t use calculator)
3. sinA = 3/ 5
(opp side is 3, hypot is 5)
cos A = ⅘
(see qu. 2 above where
opposite side and hypot were 3 and 5, adjacent
side was 4)
tanA = ¾
Use the given trigonometric ratio to
find exact values for sinA, cosA or tan
A (don’t use calculator)
4. cosA = ½
(adj. side is 1, hypot is 1)
sin A = √3 / 2
(for 30-60-90 triangle,
adjacent side is 1, hypotenuse is 2 and
opposite leg is √3)
tanA = √3 / 1 = √3
Use the given trigonometric ratio to
find exact values for sinA, cosA or tan
A (don’t use calculator)
5. tanA = 1
(opp = adjacent, must be
45-45-90 triangle, hypotenuse is√2)
sin A = 1/√2 = √2/2
cosA = √2 /2
Use the given trigonometric ratio to
find exact values for sinA, cosA or tan
A (don’t use calculator)
6. cosA = 20/29
(adj = 20, hypot = 29)
292 = 202 + opp2
Opp = √841 - 400 = 21
sin A = 21/29
tanA = 21/20