Trigonometry-Identities

Trigonometry-Identities & Equations-Inequalities

When the standard form of inequality such as R(x) > 0 or R(x) < 0 consists of 1 or more trigonometric functions of the variable arc x, then the inequality is a trigonometric inequality. Some of the examples of the trigonometric inequalities are,

sinx>a, cosx<a, tanx≤a, cotx≥a, etc.

Where the value of a is any real number, and x is an unknown variable.

Solving the inequality is to find the value of the variable arc x which satisfies the trigonometric inequality. In simple words, to find the measure of the angle that satisfies the inequality.

Trigonometric unit circle

A circle with a unit radius whose center lies at the origin O is known as the trigonometric unit circle. 

The rotation of the variable arc of this trigonometric unit circle in the anticlockwise direction gives us the basic trigonometric functions, namely:

sin x 

cos x

tan x

cot x

Cosec x

Sec x

We study a trigonometric unit circle because it helps us solve the trigonometric inequalities with proof.

The steps to find the solution for trigonometric inequalities

The inequality is converted into a trigonometric equation with the help of an equality sign; therefore, we replace the inequality sign with an equality sign.

The equation that is formed will be solved, and the value of the angle will be found, which lies between 0 to 2𝝅(0° to 360°).

If the angle lies beyond 𝝅(180°), then the value of the angle is converted to an equivalent negative value. This is done because the values repeat on the negative side of the origin.

A base interval is developed between the two values.

In case the function is asymptotes within the developed interval, then the angle value at which the function is asymptotes limits the value of the basic interval.

Finally, the solution we obtained is generalized.

Inequalities of Sin x

The inequalities of the sinx sinx>a, sin<a, sinx≤a, sinx≥a.

Sin⁡x>a

If |a|≥1, the inequality sin⁡x>has no solutions: x∈∅.

If a<−1, the solution of the inequality sin⁡x>a is any real number: x∈R.

For −1≤a<1, the solution of the inequality sin⁡x>a is expressed in the form

arc sina+2𝝅n<x<𝝅 – arc sina+2𝝅n, n∈Z

Sin⁡ x ≥ a

If a > 1, the inequality sin⁡x≥a has no solutions: x∈∅.

If a ≤ −1, the solution of the inequality sin⁡x≥a is any real number: x∈R.

Case a=1:   x = 𝝅/2 + 2𝝅n, n∈Z.

For −1 < a < 1, the solution of the non-strict inequality sin⁡x≥a includes the boundary angles and has the form

arcsin⁡a+2𝝅n ≤ x ≤ π−arcsin⁡a+2𝝅n, n∈Z.

Sin⁡ x < a

If a > 1, the solution of the inequality sin⁡x

If a ≤ −1, the inequality sin⁡x

For −1 < a ≤ 1, the solution of the inequality sin ⁡x < a lies in the interval

−𝝅−arcsin ⁡a + 2𝝅n < x < arcsin ⁡a + 2𝝅n, n∈Z.

Sin⁡ x ≤ a

If a ≥ 1, the solution of the inequality sin⁡x≤a is any real number: x∈R.

If a < −1, the inequality sin⁡x≤a has no solutions: x∈∅.

Case 

a = −1:

x = −(𝝅/2)+2πn, n∈Z.

For −1 < a < 1, the solution of the non-strict inequality sin⁡x≤a is in the interval

−𝝅−arcsin⁡ a + 2𝝅n ≤ x ≤ arcsin ⁡a + 2πn, n∈Z.

Inequalities of cos ⁡x

Cos⁡ x > a

If a ≥ 1, the inequality cos⁡ x > a has no solutions: x∈∅.

If a<−1, the solution of the inequality cos ⁡x > a is any real number: x∈R.

For −1≤ a < 1, the solution of the inequality cos⁡x>a has the form

-arccos a + 2𝝅n < x < arccos a + 2𝝅n, n∈Z

Cos⁡x ≥ a

If a > 1, the inequality cos⁡x≥a has no solutions: x∈∅.

If a ≤−1, the solution of the inequality cos⁡x≥a is any real number: x∈R.

Case a = 1:

x = 2𝝅n, n∈Z.

For −1 < a < 1, the solution of the non-strict inequality cos⁡x≥a is expressed by the formula

-arccos a + 2𝝅n ≤ x ≤ arccos a + 2𝝅n, n∈Z

Cos ⁡x < a

If a > 1, the inequality cos⁡x

If a ≤ −1, the inequality cos ⁡x < a has no solutions: x∈∅.

For −1 < a ≤ 1, the solution of the inequality cos ⁡x < a is written in the form

arccos a + 2𝝅n < x < 2𝝅 – arccos a + 2𝝅n, n∈Z

Cos ⁡x ≤ a

If a ≥ 1, the solution of the inequality cos⁡ x ≤ a is any real number: x∈R.

If a < −1, the inequality cos⁡x≤a has no solutions: x∈∅.

Case a = −1:

x = 𝝅 + 2𝝅n, n∈Z.

For −1 < a < 1, the solution of the non-strict inequality cos⁡x≤a is written as

arccos a + 2𝝅n ≤ x ≤ 2𝝅-arccos a + 2𝝅n, n∈Z

Inequalities of tan⁡ x

Tan ⁡x > a

For any real value of a, the solution of the strict inequality tan⁡x>a has the form

arctan a + 𝝅n < x < (𝝅/2) + 𝝅n, n∈Z

Tan⁡ x ≥ a

For any real value of a the solution of the inequality tan⁡x≥a is expressed in the form

arctan a + 𝝅n ≤ x ≤ (𝝅/2) + 𝝅n, n∈Z

Tan⁡ x < a

For any value of a the solution of the inequality tan⁡x

-(𝝅/2) + 𝝅n < x < arctan a + 𝝅n, n∈Z

Tan⁡ x ≤ a

For any value of a, the inequality tan⁡x≤a has the following solution:

-(𝝅/2) + 𝝅n < x ≤ arctan a + 𝝅n, n∈Z

Inequalities of Cot⁡ x

Cot⁡ x > a

For any value of a the solution of the inequality cot⁡x>a has the form

𝝅n < x < arc cot a + 𝝅n, n∈Z

Cot⁡ x ≥ a

The non-strict inequality cot⁡x≥a has a similar solution:

𝝅n < x ≤ arc cot a + 𝝅n, n∈Z.

Cot⁡ x < a

For any value of a the solution of the inequality cot ⁡x < a lies on the open interval

arccot a + 𝝅n < x < 𝝅 + 𝝅n, n∈Z

Cot ⁡x ≤ a

For any value of a the solution of the non-strict inequality cot⁡ x ≤ a is in the half-open interval

arc cot a + 𝝅n ≤ x < 𝝅 + 𝝅n, n∈Z

Conclusion

Trigonometric inequalities are inequalities with trigonometric functions. Trigonometric unit circles are used to prove the value of trigonometric inequalities.