Solving typical problems on trigonometric equations
In this lesson you will find the solutions of these typical trigonometric equations:
1.
=
. 2.
=
. 3.
=
. 4.
=
. 5.
=
. 6.
=
.
Problem 1
Solve an equation
=
in the interval [
,
)
Solution
=
--->
cos^2(x) =
=
,
cos(x) = +/-
= +/-
.
cos(x) =
---> x =
or/and x =
.
cos(x) =
---> x =
or/and x =
.
Answer. x =
,
,
and
.
Problem 2
Solve an equation
=
in the following domain 0 <= x <
.
Solution
Problem 3
Solve an equation
=
.
Solution
= sec(x)
is the same as
+
=
.
Multiply both sides by cos(x). You will get
+
= 1.
Multiply both sides by
. You will get
+
=
.
Recall that
=
,
=
.
Therefore, you can write the last equation as
=
.
Apply the addition formula for sine. ( It is cos(a)*sin(b) + sin(a)*cos(b) = sin(a+b).
See the lesson Addition and subtraction formulas in this site ). You will get
=
.
It implies
=
or
=
.
Hence, x =
=
or x =
=
=
.
The last root doesn't fit due to "sec" in the original equation.
Answer. x =
, or
for any integer "k".
Problem 4
Solve for x:
=
.
Solution
sin(x) + cos(x) = -1. (1) (It is the original equation)
Square its both sides. You will get
=
. (2)
From the other side, there is an identity
==
. (3)
Comparing (2) and (3), you get
2*sin(x)*cos(x) = 0, or sin(x)*cos(x) = 0. (4)
Equation (4) splits in two independent equations
1) sin(x) = 0 ---> x =
, k = 0. +/-1. +/-2, . . . (5)
2) cos(x) = 0 ---> x =
, k = 0. +/-1. +/-2, . . . (6)
Now we should check which of the found values (5), (6) satisfy the original equation.
Of the set (5), all x satisfy sin(x) = 0. Hence, only those of (5) satisfy the original equation where cos(x) = -1.
They are x =
, n = 0, +/-1. +/-2, . . . , or
x =
, n = 0, +/-1. +/-2, . . . , (5').
Of the set (6), all x satisfy cos(x) = 0. Hence, only those of (6) satisfy the original equation where sin(x) = -1.
They are x =
, n = 0, +/-1. +/-2, . . . , or
x =
, n = 0, +/-1. +/-2, . . . , (6').
Answer. The union of the sets (5') and (6') is the solution of the original equation.
Problem 5
Solve an equation
=
.
Solution
=
.
Multiply both sides by
. You will get
= 1. (1)
Notice and use that
=
=
.
Then from (1) you will get
= 1. (2)
Now use the formula sin(a)*cos(b) - cos(a)*sin(b) = sin(a-b). Then from (2) you will get
= 1.
It implies
=
.
Then
=
=
.
It is the same as
=
in the interval 0 <=
<
.
The plot below confirms the solution (
~= 2.33 )
Plots y =
and y =
Problem 6
Find the general solution to an equation
=
.
Solution
=
.
Multiply both sides by
. You will get
=
. (1)
Notice that
=
,
=
.
Substitute it into the left side of (1). You will get
=
. (2)
Apply the formula sin(a)*cos(b) + cos(a)*sin(b) = sin(a+b) to the left side of (2). You ill get
=
. (3)
It implies
=
, k = 0, +/-1, +/-2, . . . or
=
, k = 0, +/-1, +/-2, . . .
Thus there are two sets of solutions:
1. x =
=
, which is equivalent to x =
,
and the other family
2. x =
=
Answer. There are two sets of solutions: 1) x =
and 2) x =
, k = 0, +/-1, +/-2, . . .
The plot below confirms these solutions.
Plots y =
and y = 1
My other lessons on calculating trig functions and solving trig equations in this site are
- Calculating trigonometric functions of angles
- Advanced problems on calculating trigonometric functions of angles
- Evaluating trigonometric expressions
- Solve these trigonometry problems without using a calculator
- Finding the slope of the bisector to the angle formed by two given lines in a coordinate plane
- Solving simple problems on trigonometric equations
- Solving more complicated problems on trigonometric equations
- Solving advanced problems on trigonometric equations
- Challenging problems on trigonometric equations
- Miscellaneous problems on solving trigonometric equations
- Solving twisted trigonometric equations
- Truly elegant solution to one trigonometric equation
- Non-standard Trigonometry problems
- Proving Trigonometry identities
- Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)
- Find the height
- Word problems on Trigonometric functions
- Solving upper-league Trigonometry equations
- Math OLYMPIAD level problems on Trigonometry
- Trigonometry entertainment problems
- OVERVIEW of lessons on calculating trig functions and solving trig equations
Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.