Download Values of Trigonometric Functions (Sec. 6.4)

Values of Trigonometric Functions (Sec. 6.4)
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Details / Examples
How are we really going to
find values of trig
functions? (And why
bother with reference
angles?)
(1) “I know this angle is over 90°, but the stupid calculator
keeps saying sin-1(.636) is 39.494°!”
(2) “Mr. Byrd says he wants the exact value for tan(82π/6), and
I have a feeling –1.732050808 isn’t exact!”
Definition of reference angle Let θ be a nonquadrantal angle in standard position. The
Review terms: nonquadrantal
angle, terminal side
Reference angles in each
quadrant
reference angle for θ is the acute angle θR that the terminal side
of θ makes with the x-axis.
With 0° < θ < 360° or 0 < θ < 2π:
If θ is greater than 360° or less than 0°, first find the
_____________ angle θ with 0° < θ < 360° or 0 < θ < 2π.
Example: finding reference Find the reference angle θR for θ, and sketch both angles in
standard position.
angles
(a) θ = 315°
(b) θ = -240°
(c) θ = 5π/6
(d) θ = 4
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Theorem on Reference
Angles
If θ is a _______________ angle in standard position, then to
find the value of a trigonometric function at θ, find its value for
the _____________________ and prefix the appropriate sign.
But what is the appropriate sign?
Signs of Trigonometric
Functions
Functions that are positive in the quadrant shown:
Mnemonic: “A Smart Trig
Class” = All, Sin, Tan, Cos*
II
I
Sin, csc
ALL
III
IV
Tan, cot
Cos, sec
.
*Functions have the _______ sign as their reciprocal, so the
mnemonic mentions only sin, tan, cos.
Example: using reference
angles
Find the exact values of sin θ, cos θ, and tan θ if
Finding the angle from the
function value
A common real-life situation: hypotenuse & height are known.
Solution: inverse trigonometric functions.
Example
If sin θ = 0.6635, what is θ?
(a) θ = 5π/6
(b) θ = 315°
On calculator, find approx. value of sin-1 0.6635
Finding acute angle
solutions of equations
with a calculator
“I know this angle is over 90°, but the stupid calculator keeps
saying sin-1(.636) is 39.494°!”
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Finding angles with a
calculator
Note: if cos θ is negative, θ is
not the negative of the
reference angle!
DAB, March 2011
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