Download Mod 2 - Aim #27 - Manhasset Public Schools

CC Geometry H
Aim #27: Students rewrite the Pythagorean theorem in terms of sine and cosine
ratios and write tangent as an identity in terms of sine and cosine.
Do Now: 1) In a right triangle, with acute angle θ, sin θ = 1/2. What is the
degree measure of θ? What is the value of cos θ?
θ
2) In a right triangle, with acute angle θ, sin θ = 7/9. What is the value of tan θ?
θ
3) What common Pythagorean triple was probably modeled in the construction of
o
the triangle in figure 2? Use sin 53 ≈ 0.8.
The Pythagorean Identity
-Let θ be the angle such that sin θ = 0.8. Above, we used an approximation. We do
not know exactly what the angle measure is, but we will assign the angle measure
that results in the sine of the angle as 0.8 the label θ. We have the following
triangle:
0.8
1
θ
0.6
-Since the value of sin θ is 0.8 and the value of cos θ is 0.6, how can we rewrite
the leg lengths?
1
θ
Apply the Pythagorean Theorem to this triangle:
sin θ
1
θ
cos θ
The statement above is called a Pythagorean Identity.
To prove this identity, we will use this diagram:
2
2
2
opp + adj = hyp
opp
hyp
θ
2
Divide both sides by hyp :
adj
The Quotient Identity
sin θ
Prove that tan θ =
cos θ
sin θ =
c
a
cos θ =
θ
b
tan θ =
If you are given one of the values of sin θ, cos θ or tan θ, we can find the other
two values using the identities (sin θ)2 + (cos θ)2 = 1 and tan θ = sin θ or by using
cos θ
the Pythagorean theorem.
Applying the trigonometric identities:
1) In a right triangle, with acute angle θ, sin θ = 1/2. Use the Pythagorean identity
to determine the value of cos θ.
2) In a right triangle, with acute angle θ, sin θ = 7/9. Use the trigonometric
identities to determine the value of tan θ.
3) If cos β = 2/3, use the trigonometric identities to find sin β and tan β.
4) Find the missing side lengths of the following triangle using sine, cosine, and/or
tangent. Round your answer to the nearest ten-thousandth.
5) The right triangle shown is taken from a slice of a right rectangular pyramid
with a square base.
a. Find the height of the pyramid to the nearest tenth.
b. Find the lengths of the sides of the base of
the pyramid to the nearest tenth.
c. Find the lateral surface area of the right rectangular pyramid to the nearest
tenth. (lateral surface area: the area of the side faces, not including the base.)
6) A machinist is fabricating a wedge in the shape of a right triangular prism. One
acute angle of the right triangular base is 33, and the opposite side is 6.5 cm. Find
the length of the edges labeled l and m using sine, cosine, and/or tangent. Round
your answer to the nearest thousandth of a centimeter.
Let's Sum It Up
• The Pythagorean Identity is (sin θ) + (cos θ) = 1
opp
sin θ
• The Quotient Identity can be represented as tan θ =
or tan θ = adj
cos θ
2
2
• If you have one of the values of sin θ, cos θ, or tan θ, you can use the
above identities or the Pythagorean Theorem.
Name_____________________
Date _____________________
1) If cos β =
2) If sin θ =
CC Geometry H
HW #27
, use trigonometric identities to find sin β and tan β.
, use trigonometric identities to find cos θ and tan θ.
3) If tan θ = 5, use the Pythagorean theorem to find the hypotenuse, and then find
cos θ and sin θ.
θ
4) If sin θ =
, use trigonometric identities to find cos θ and tan θ.
5) Find the missing side lengths of the following triangle using sine, cosine, and/or
tangent. Round your answer to the nearest ten-thousandth.
OVER
6) A surveying crew has two points A and B marked along a roadside at a distance of
400 yd. A third point C is marked at the back corner of a property along a
perpendicular to the road at B. A straight path joining C to A forms a 28 degree angle
with the road. Find the distance from the road to point C at the back of the property
and the distance from A to C using sine, cosine, and/or tangent. Round your answer to
the nearest thousandth.
Review:
1) In quadrilateral ABCD, the diagonals bisect its angles. If the diagonals are not
congruent, quadrilateral ABCD must be a _______________.
2) In the diagram below, QM is a median of triangle PQR and point C is the
centroid of triangle PQR. If QC = 5x and CM = x + 12, determine and state the
length of QM.
3) Use a compass and straightedge to divide line segment AB below into three
congruent parts. [Leave all construction marks.]