Warm up
Right Triangle Trigonometry
Objective To learn the trigonometric
functions and how they apply to a
right triangle.
The Trigonometric Functions
we will be looking at
SINE
COSINE
TANGENT
The Trigonometric Functions
SINE
COSINE
TANGENT
Greek Letter q
Prounounced
“theta”
Represents an unknown angle
Opp
sin 
Hyp
hypotenuse
Adj
cos 
Hyp
Opp
tan 
Adj
q
adjacent
opposite
opposite
Finding sin, cos, and tan
SOHCAHTOA
Opp
sin q 
Hyp
Adj
cos q 
Hyp
8
10
4

5
10
8
3
6

10
5
q
Opp
tan q 
Adj
4
8

6
3
6
Find the sine, the cosine, and the tangent of angle A.
Give a fraction and decimal answer (round to 4 places).
10.8
9
A
9
opp

sin A 
hypo 10.8  .8333
adj
6
cos A 

hypo 10.8
 .5555
6
opp
tan A 
adj
9

6
 1.5
Find the values of the three trigonometric functions of q.
?
5
4
q
Pythagorean Theorem:
(3)² + (4)² = c²
5=c
3
opp 4
adj 3
opp 4

sin q 


cos q 
tan q 
hyp 5
hyp 5
adj
3
Sine
• Find the sin of α
β
8

α
C
10
A
Find the sine, the cosine, and the tangent of angle A
B
Give a fraction and
decimal answer (round
to 4 decimal places).
24.5
8.2
A
23.1
opp  8.2
sin A 
 .3347
24
.
5
hypo
23.1
adj

cos A 
24.5  .9429
hypo
opp
tan A 
adj
8 .2

23.1  .3550
Cosine
• Find the cosine and tan of α
β
6
5

α
C
√11
A
The Reciprocal Trigonometric
Ratios
• Often it is useful to use the reciprocal ratios,
depending on the problem.
– Cosecant θ is the reciprocal of sine θ,
– Secant θ is the reciprocal of cosine θ, and
– Cotangent θ is the reciprocal of tangent θ
The Reciprocal Trigonometric
Ratios
hyp
opp
opp
sin  
hyp
hyp
sec  
adj
adj
cos  
hyp
adj
cot  
opp
opp
tan  
adj
csc  
Trigonometric Identities
1
1
sin x 
csc x 
csc x
sin x
1
cos x 
sec x
1
tan x 
cot x
1
sec x 
cos x
1
cot x 
tan x
Examples
Find sec, csc, and cot for
angle θ
1
1
2
θ
3
2
Special Angles
• Special Right Triangles are 30-60-90 and 45-4590
45
60
o
1
2
1
√3
3
0o
o
√2
45
1
o
Fill in the Chart
θ in degrees
sin θ
cos θ
tan θ
csc θ
sec θ
cot θ
30
45
60
θ in degrees
30
45
60
Relationship between Sine and
Cosine
• sin (α) = cos (  )
β
5
3

α
C
4
A
Cofunctions
Cofunctions of complementary angles are
____________
. If θ is an acute angle, then:
equal

cosq
 q )  __________
cos(90  q )  __________
sin q

cot q
 q )  __________
cot(90  q )  __________
tan q
sin(90
tan(90
q
sec(90  q )  csc
__________



csc(90  q )  __________
secq

Relationship between Sine and
Cosine
• Look at the Pythagorean Theorem
• (adj)2 + (opp)2 = (hyp)2
• Divide each side by (hyp)2
•
•
•
•
(adj)2 + (opp)2 = (hyp)2
(hyp)2 (hyp)2 (hyp)2
(adj)2 + (opp)2 = 1
(hyp)2 (hyp)2
Relationship between Sine and
Cosine

•
(sin (x))2 + (cos (x))2 = 1
•
sin2(x) + cos2(x) = 1
Sources
• lhsblogs.typepad.com/files/section_5.2_rig
ht-triangle-trigonometry-2.ppt, Oct.1,
2013