Trigonometry Examples

Solve the System of @WORD sin(x)<0 , sec(x)>0
,
Step 1
Solve for .
Tap for more steps...
Step 1.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 1.2
Simplify the right side.
Tap for more steps...
Step 1.2.1
The exact value of is .
Step 1.3
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 1.4
Subtract from .
Step 1.5
Find the period of .
Tap for more steps...
Step 1.5.1
The period of the function can be calculated using .
Step 1.5.2
Replace with in the formula for period.
Step 1.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.5.4
Divide by .
Step 1.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 1.7
Consolidate the answers.
, for any integer
Step 1.8
Use each root to create test intervals.
Step 1.9
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Tap for more steps...
Step 1.9.1
Test a value on the interval to see if it makes the inequality true.
Tap for more steps...
Step 1.9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.9.1.2
Replace with in the original inequality.
Step 1.9.1.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 1.9.2
Test a value on the interval to see if it makes the inequality true.
Tap for more steps...
Step 1.9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.9.2.2
Replace with in the original inequality.
Step 1.9.2.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 1.9.3
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
True
Step 1.10
The solution consists of all of the true intervals.
, for any integer
, for any integer
Step 2
The range of secant is and . Since does not fall in this range, there is no solution.
No solution