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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 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
the triangle in figure 2? Use sin 53o ≈ 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:
1
sin θ
θ
cos θ
The statement above is called the Pythagorean Identity.
To prove this identity, we will use this diagram:
opp2 + adj2 = hyp2
hyp
opp
θ
Divide both sides by hyp2:
adj
The trigonometric 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
θ
two values using the identities (sin θ)2 + (cos θ)2 = 1 and tan θ = sin
or by using
cos θ
the Pythagorean theorem.
Applying the Pythagorean Identity:
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 Pythagorean identity
to determine the value of tan θ.
3) If cos β = 2/3, use the Pythagorean identity and the trigonometric identity 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. (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 33o, 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.
adj
Let's Sum It Up
• The Pythagorean Identity is (sin θ)2 + (cos θ)2 = 1.
opp
• Tangent can be represented as tan θ = sin θ
or tan θ =
adj
cos θ
.
• 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 four
congruent parts. [Leave all construction marks.]