-
Home
- Algebra and trigonometry
- The unit circle: sine and cosine
- The other trigonometric functions
Key equations
Tangent function |
|
Secant function |
|
Cosecant function |
|
Cotangent function |
|
Key concepts
- The tangent of an angle is the ratio of the
y -value to the
x -value of the corresponding point on the unit circle.
- The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
- The six trigonometric functions can be found from a point on the unit circle. See
[link]
.
- Trigonometric functions can also be found from an angle. See
[link] .
- Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See
[link] .
- A function is said to be even if
and odd if
for all
x in the domain of
f.
- Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
- Even and odd properties can be used to evaluate trigonometric functions. See
[link] .
- The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
- Identities can be used to evaluate trigonometric functions. See
[link] and
[link]
.
- Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See
[link] .The trigonometric functions repeat at regular intervals.
- The period
of a repeating function
is the smallest interval such that
for any value of
- The values of trigonometric functions can be found by mathematical analysis. See
[link] and
[link]
.
- To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See
[link] .
Section exercises
Verbal
On an interval of
can the sine and cosine values of a radian measure ever be equal? If so, where?
Yes, when the reference angle is
and the terminal side of the angle is in quadrants I and III. Thus, a
the sine and cosine values are equal.
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For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?
Substitute the sine of the angle in for
in the Pythagorean Theorem
Solve for
and take the negative solution.
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Tangent and cotangent have a period of
What does this tell us about the output of these functions?
The outputs of tangent and cotangent will repeat every
units.
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Algebraic
For the following exercises, find the exact value of each expression.
For the following exercises, use reference angles to evaluate the expression.
If
and
is in quadrant III, find
and
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Questions & Answers
(Pcos∅+qsin∅)/(pcos∅-psin∅)
how to answer the activity
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
If
, ,
are the roots of the equation
3 2 0,
x px qx r
Find the value of
1
.
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
lairenea's age is 23yrs
ACKA
Laurene is 46 yrs and Mae is 23 is
Solomon
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0
(-π<A<=π
create a lesson plan about this lesson
Excusme but what are you wrot?
Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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