The study of triangles Relationship between sides and angles of a right triangle › What is…
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Introduction to Trigonometry
Basic Trigonometric Functions
What is Trigonometry? The study of triangles Relationship between sides and angles of a right triangle
› What is a right triangle? A triangle with a 90⁰ angle
90°
Review Right Triangles In relation to angle a, the
sides of the triangle are:• hypotenuse - always
longest side and side across from right angle (90⁰)
• adjacent - side closest to
angle a• opposite - side opposite
to angle a
hypotenus
e
a
adja
cent
opposite
90⁰
Review Right Triangles
Label the sides for angle b:
• hypotenuse• adjacent• opposite
b
?
?
?hypotenus
e
opposite
adjacent
90°
Trigonometric FunctionsRatios of the sides in relation to angle a:
sine cosine tangent
hypotenus
e a
adja
cent
opposite
90°
Trigonometric Functions:SINE
abbreviation: sin
sin(a)=› 0 ≤ sin ≤ 1
Example: sin(60°)= = ~.866
Ratio of opposite side to hypotenuse for 60° angle is to 2 (.866 to 1)
hypotenus
e a
adja
cent
opposite
90°
oppositehypotenus
e √32
Trigonometric Functions:COSINE
abbreviation: cos
cos(a)=› 0 ≤ cos ≤ 1
Example: cos(60°)= = .5
Ratio of adjacent side to hypotenuse for 60° angle is 1 to 2 (half)
hypotenus
e a
adja
cent
opposite
90°
adjacenthypotenus
e12
Trigonometric Functions:TANGENTabbreviation: tan
tan(a)=› 0 ≤ tan ≤
Example: tan(60°)= = ~1.732
Ratio of opposite side to adjacent side for 60° angle is to 1 (1.732 to 1)
hypotenus
e a
adja
cent
opposite
90°
oppositeadjacent∞oppositeadjacent
Trigonometric Functions:REMEMBER:
Sine =
Cosine =
Tangent =
hypotenus
e a
adja
cent
opposite
90°
Opposite
AdjacentHypotenuse
Hypotenuse
Adjacent
OppositeSOH – CAH – TOASOH TOACAH
Using Trigonometric Functions:
For any right triangle:calculate other sides if one side and angle known
calculate angle if two sides known
90°
Calculating Sides:One Side and Angle Known
What is known?• angle (50°) and adjacent side (2)
Solving for hypotenuse: Which function uses adjacent and hypotenuse?
hypotenus
e 50°
2
opposite
90°
COSINE
Calculating Sides:One Side and Angle Known
What is known?• angle (50°) and adjacent side (2)
Solving for hypotenuse: cos(50°)= =
hypotenuse = ~3.111
hypotenus
e 50°
2
opposite
90°
2 hypotenuse
3.111
~0.643
Calculating Sides:One Side and Angle Known
Now we know:• angle (50°) and hypotenuse (3.111)
Solving for opposite: Which function uses opposite and hypotenuse?3.111
50°
2
opposite
90°
SINE
Calculating Sides:One Side and Angle Known
Now we know:• angle (50°) and hypotenuse
(3.111)Solving for opposite:
sin(50°)= = opposite = ~2.384
3.111
50°
2
opposite
90°
opposite3.111
2.384
~.766
Calculating Angle:Two Sides KnownWhat is known?• adjacent (3) and opposite (5)
Solving for angle (a): Which function uses
adjacent and opposite?
hypotenus
e a
3
5
90°
TANGENT
Calculating Angle:Two Sides KnownWhat is known?• adjacent (3) and opposite (5)
Solving for angle (a): tan(a)= =* need to use inverse tan → tan-1(.6) = a =
~30.964°
hypotenus
e
3
5
90°
35
a 30.964° .6
TEKS Reference§111.35. Precalculus (c) Knowledge and skills.(3) The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to:(A) investigate properties of trigonometric and polynomial functions;
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