P.o.D. – Evaluate each trigonometric
function.
1.) sin
3𝜋
2
2.) cos −
3.) tan −
8𝜋
3
9𝜋
4
1.) -1
2.) – ½
3.) -1
4.3 – Right Triangle Trigonometry
Learning Target: be able to use
trigonometric functions to model and
solve real life problems.
Consider the Right Triangle:
Hypotenuse
Side opposite angle A
Side Adjacent to angle A
Right Triangle Definitions:
𝑜𝑝𝑝
𝑜𝑝𝑝
𝑎𝑑𝑗
tan 𝜃 =
sin 𝜃 =
cos
𝜃
=
𝑎𝑑𝑗
ℎ𝑦𝑝
ℎ𝑦𝑝
ℎ𝑦𝑝
ℎ𝑦𝑝
𝑎𝑑𝑗
csc 𝜃 =
sec 𝜃 =
cot 𝜃 =
𝑜𝑝𝑝
𝑎𝑑𝑗
𝑜𝑝𝑝
Some Mnemonic Devices:
“Some Old Hippie Caught Another
Hippie Tripping On Acid”
S
OH
C T
AH
OA
Oscar Has A Hairy Old Armpit.
Old Houses Are Homes Of Angels.
4
2
𝜃
x
EX: Find the value of the six
trigonometric functions of 𝜃 as shown in
the figure above.
First, find the length of the missing side,
x.
𝑥 2 + 22 = 42 → 𝑥 2 = 16 − 4 = 12
→ 𝑥 2 = √12 = 2√3
Now evaluate each trig function.
4
2 1
csc 𝜃 = = 2
sin 𝜃 = =
2
4 2
2√3 √3
2
2√3
cos 𝜃 =
=
sec 𝜃 =
=
4
2
3
√3
2
1
√3
tan 𝜃 =
=
cot 𝜃 =
= √3
2√3 √3
1
√3
=
3
Properties of a 45/45/90 Triangle:
45°
x√2
x
45°
x
EX: Find the value of cot 45°, sec 45°,
and csc 45°.
𝑥
cot 45° = = 1
𝑥
ℎ𝑦𝑝 𝑥 √2
sec 45° =
=
= √2
𝑎𝑑𝑗
𝑥
ℎ𝑦𝑝 𝑥 √2
csc 45° =
=
= √2
𝑜𝑝𝑝
𝑥
Properties of the 30/60/90 Triangle:
60°
2x
x
30°
x√3
EX: Use the properties above to find the
values of tan 60°, cot 60°, tan 30°, and
cot 30°.
𝑜𝑝𝑝 𝑥 √3
tan 60° =
=
= √3
𝑎𝑑𝑗
𝑥
𝑎𝑑𝑗
𝑥
1
√3
cot 60° =
=
=
=
𝑜𝑝𝑝 𝑥 √3 √3
3
𝑥
√3
tan 30° =
=
3
𝑥 √3
𝑥 √3
cot 30° =
= √3
𝑥
*We should notice something important
about complementary angles and their
trig functions.
tan 60° = cot 30°
*Memorize the Sines, Cosines, and
Tangents of Special Angles on page 303.
𝜋 1
sin 30° = sin =
6 2
𝜋
4
√2
=
2
𝜋
sin 60° = sin
3
√3
=
2
sin 45° = sin
𝜋
6
√3
=
2
𝜋
cos 45° = cos
4
√2
=
2
𝜋
cos 60° = cos
3
1
=
2
cos 30° = cos
𝜋
6
√3
=
3
𝜋
tan 45° = tan
4
=1
tan 30° = tan
𝜋
tan 60° = tan
3
= √3
*Memorize the Cofunctions of
Complementary Angles:
sin(90° − 𝜃)
cos(90° − 𝜃)
= cos 𝜃
= sin 𝜃
tan(90° − 𝜃)
cot(90° − 𝜃)
= cot 𝜃
= tan 𝜃
sec(90° − 𝜃)
csc(90° − 𝜃)
= csc 𝜃
= sec 𝜃
*Memorize the Fundamental
Trigonometric Identities:
Reciprocal Identities:
sin 𝜃 =
cos 𝜃 =
tan 𝜃 =
1
csc 𝜃
1
sec 𝜃
1
cot 𝜃
or csc 𝜃 =
or sec 𝜃 =
or cot 𝜃 =
Quotient Identities:
1
sin 𝜃
1
cos 𝜃
1
tan 𝜃
tan 𝜃 =
sin 𝜃
cos 𝜃
or cot 𝜃 =
cos 𝜃
sin 𝜃
Pythagorean Identities:
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃
1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃
EX: Let 𝜃 be an acute angle such that
cos 𝜃 = 0.96. Find the values of sin 𝜃
and tan 𝜃 using trigonometric identities.
Use a Pythagorean identity to find sine.
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 →
𝑠𝑖𝑛2 𝜃 + (0.96)2 = 1 →
𝑠𝑖𝑛2 𝜃 = 1 − (0.96)2 = .0784
→ sin 𝜃 = √. 0784 = 0.28
We can use a Reciprocal Identity and a
Pythagorean Identity to find tangent.
1
1
25
sec 𝜃 =
=
=
cos 𝜃 . 96 24
1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃 →
2
25
1 + 𝑡𝑎𝑛 𝜃 = ( ) →
24
2
2
25
𝑡𝑎𝑛2 𝜃 = ( ) − 1 →
24
49
49
𝑡𝑎𝑛 𝜃 =
→ tan 𝜃 = √
576
576
2
≈ 0.2916667
*We could also solve this problem as a
Right Triangle.
EX: Let 𝛽 be an acute angle such that
tan 𝛽 = 4. Find the values of cot 𝛽 and
sec 𝛽 using trigonometric identities.
We can find cotangent by using a
reciprocal identity.
1
1
cot 𝛽 =
=
tan 𝛽 4
We can use a Pythagorean Identity to
find secant.
1 + 𝑡𝑎𝑛2 𝛽 = 𝑠𝑒𝑐 2 𝛽 →
1 + (4)2 = 𝑠𝑒𝑐 2 𝛽 →
17 = 𝑠𝑒𝑐 2 𝛽 →
√17 = sec 𝛽
EX: Use a calculator to evaluate
csc(34°30′ 36′′ ).
*Be sure that your calculator is in
DEGREE mode.
Avoiding Common Errors:
Always check to be sure that your
calculator is set to the correct mode
(radian or degree) before evaluating a
trig function.
EX: A biologist wants to know the width
W of a river in order to properly set
instruments for studying the pollutants
in the water. From a point A, the
biologist walks downstream 70 feet and
sights to point C. From this sighting, it is
determined that 𝜃 = 54°. How wide is
the river?
Always begin by drawing a picture.
(Draw on the whiteboard).
𝑊
tan 54° =
70
→ 70 tan 54° = 𝑊 → 𝑊
= 96.347 𝑓𝑡
EX: A 12-meter flagpole casts a 12-meter
shadow. Find 𝜃, the angle of elevation to
the sun.
Begin by drawing a picture. (on
whiteboard)
12
tan 𝜃 =
=1
12
Because this is a 45/45/90 triangle, we
should have this angle memorized. We
should recognize that 𝜃 = 45°.
To solve for an angle, we must use an
arc function:
𝑎𝑟𝑐𝑠𝑖𝑛𝑒 = 𝑠𝑖𝑛−1
𝑎𝑟𝑐𝑐𝑜𝑠𝑖𝑛𝑒 = 𝑐𝑜𝑠 −1
𝑎𝑟𝑐𝑡𝑎𝑛𝑔𝑒𝑛𝑡 = 𝑡𝑎𝑛−1
EX: A ramp 17 ½ feet in length rises to a
loading platform that is 3 ½ feet off the
ground. Find the angle 𝜃 that the ramp
makes with the ground.
Draw a picture on the whiteboard.
3.5
3.5
−1
sin 𝜃 =
→ 𝜃 = 𝑠𝑖𝑛 (
)
17.5
17.5
= 11.537°
http://www.youtube.com/watch?v=LE6
dmczMc68
Upon completion of this lesson, you
should be able to:
1. Solve right triangles.
2. Explain the properties of a 45-45-90
and 30-60-90 triangle.
3. Recite common trig values.
4. Identify the reciprocal, quotient,
and Pythagorean identities and
apply them.
For more information, visit
http://www.themathpage.com/atrig/solve-righttriangles.htm
HW Pg.308
83-85
3-66 3rds, 67-68, 73-78,