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Name____________________________________
9-1 Notes
Geometry
Date________
Lesson 9-1: Solving for a Side of a Right Triangle Using Trigonometry
Learning Goal: What are trigonometric ratios and how can we use them to solve for a side?
Example 1) Solve for the missing side in the right triangle shown below.
What’s your thinking?
Flashback!
Find 𝑓(3) given 𝑓(𝑥) = 2𝑥
The function is acting on the “input” to give us an “output”.
Input
Output
In this unit, we will be discussing three new functions that stem from special relationships within right triangles, so
we call then trigonometric functions. Can you find them on your calculator?
These functions are special because they only act on angle measures and their values are found by using specific ratios.
These ratios are formed by using the
lengths of the right triangles.
𝑓(𝑥)
sin ( 𝑥) where x is the measure of an angle
𝑘(𝑥)
cos (𝑥) where x is the measure of an angle
𝑔(𝑥)
tan (𝑥)
where x is the measure of an angle
These functions will help us solve for missing angles
and missing sides of right triangles when
Pythagorean Theorem can’t help us.
Transfer your knowledge…
If we see 𝐜𝐨𝐬(𝟔𝟎) then we know that the function ________________ is acting on _____________
to give us some type of output which we call the VALUE of the function.
What will be the output/ VALUE of the Trig Function?
To find the output/ value of a trig function, we use ratios! But first, let’s discuss angle-side relationships.
Angle-Side Relationships
Opposite Side
The side ___________________ from a given angle
Adjacent Side
The side ___________________ a given angle * NOT the _____________________
Hypotenuse
The side ___________________ the _______________________
Let's Practice
1) What is the length of the side opposite from angle A? ________
**Look @ how the sides are labeled
here!
2) What is the length of the side adjacent to angle B? _________
Where is side ‘a’ located?
3) What is the length of the hypotenuse? __________
Sides of triangles are sometimes
labeled by the angle
_________________ them!
When calculating the values of the functions, we will always use these special ratios!
Trigonometric Ratios
Or,…
SOH – CAH – TOA
Let’s practice writing trig ratios!
1) The diagram right, shows right triangle XYZ.
Write the ratio that represents...
a) the sine of ∠X
b) the cosine of ∠X
c) the tangent of ∠Y
d) the cosine of ∠Y
CHECK YOUR CALCULATOR MODE
Solving for a Side Using Trigonometric Ratios
2) In right triangle ABC, hypotenuse AB=15 and angle A=35º. Find leg length, BC, to the nearest tenth.
Why CAN’T we use Pythagorean Theorem here??
Let’s try another!
3) In right triangle ABC, leg length BC=20 and angle B = 41˚. Find hypotenuse length BA to the nearest hundredth.
SOHCAHTOA Partner Playtime
Choose who will be partner peanut butter and who will be partner jelly. Work on your column’s questions. Each problem is different,
but your multiple choice answers should match. If there is a discrepancy between answers, figure out who is right and help your
partner correct their mistake. Work on the last 2 problems together
1
Partner Peanut Butter
below, the measure of
, and
.
Which ratio represents the sine of
?
In
1)
,
,
Partner Jelly
The diagram below shows right triangle UPC.
Which ratio represents the sine of
1)
3)
3)
2)
2)
2
3
4)
Which ratio represents
diagram of
1)
3)
1)
3)
2)
4)
2)
4)
In the accompanying diagram of right triangle ABC,
,
,
, and
.
2)
?
?
3)
4)
4)
In triangle MCT, the measure of
,
, and
Which ratio represents the sine of
1)
3)
2)
in the accompanying
The diagram below shows right triangle ABC.
Which ratio represents the tangent of
?
1)
3)
2)
4
4)
Which ratio represents the cosine of angle A in the
right triangle below?
What is
1)
?
4)
,
?
In
, the measure of
and
.
,
Which ratio represents the tangent of
1)
3)
2)
4)
,
?
5
Partner Peanut Butter
As shown in the diagram
below, a building casts a
72 foot shadow on the
ground when the angle
of elevation of the Sun is 40°.
Partner Jelly
The accompanying diagram shows a ramp 30 feet long
leaning against a wall at a construction site.
If the ramp forms an angle
of 32° with the ground,
how high above the
ground, to the nearest
tenth, is the top of the
ramp?
How tall is the building, to the nearest foot?
6
1) 60 2) 46 3) 86 4) 94
1) 15.9 ft 2) 18.7 ft 3) 25.4 ft 4) 56.6 ft
As shown in the diagram below, the angle of elevation
from a point on the ground to the top of the
tree is 34°.
A tree casts a 25-foot shadow on a sunny day, as
shown in the diagram below.
If the point is 20 feet from the base of the tree, what is
the height of the tree, to the nearest tenth of a foot?
1) 29.7 2) 13.5 3) 16.6 4) 11.2
If the angle of elevation from the tip of the shadow to
the top of the tree is 32°, what is the height of the tree
to the nearest tenth of a foot?
1) 13.2 2) 15.6 3) 21.2 4) 40.0
Peanut Butter Jelly Time! Try these as a team now
7. A support post leans against a wall, making a 70° angle with the ground. The support post is 20 feet above the ground
where it leans along the wall. To the nearest tenth of a foot, how far along the ground is the base of the support from
the wall?
8. In right triangle ABC,
,
, and
. What is the length of the hypotenuse?
1)
2)
3) 8
4)
9. From a point on level ground 25 ft. from the base of a tower, the angle of elevation to the top of the tower is 78°, as
shown in the accompanying diagram. Find the height of the tower to the nearest tenth of a foot.
10. A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram. From a boat
located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff is 18° and the angle of
elevation to the top of the lighthouse is 28°. What is the height of the lighthouse, x, to the nearest tenth of a
foot?
Name: _________________________________________
9-1 HW
Geometry Pd. ____
Date: ______
Directions: Complete EACH of the following problems; CHECK YOUR ANSWERS!!
1) Using the diagram below find the sine, cosine and tangent ratios
of ∠ABC?
Sin: __________
Cos: _________
Tan: ________
2) A ramp is leaning against a wall. The angle the ramp makes with the ground is 18°. The length of the ramp is 12 feet.
Find the length from the bottom of the ramp to the wall to the nearest tenth.
In the diagram below, a window of a house is 15 feet above the ground. A ladder is placed against the house with its
base at an angle of 75° with the ground. Determine and state the length of the ladder to the nearest tenth of a foot.
5) The tailgate of a truck is 2 ft above the ground. The incline of a ramp used for loading the truck is 11°, as shown
below.
Find to the nearest tenth of a foot, the length of the ramp.
5
6) Sketch right triangle DEF with E the right angle. If cos<DFE = 6, determine the sin<DFE (Leave any radicals in your final
answer, NO DECIMALS).
7) The diagram shows a rectangle ABCD with diagonal BD drawn.
What is the perimeter of the rectangle (ABCD) to the nearest tenth?