Inverse Trigonometric Functions

Category: Trigonometry, Algebra

"Published in Suisun City, California, USA"

Rewrite the expression as an algebraic expression in x for 

Solution:

Consider the given equation above, we have

Let

and

so that the above equation becomes

Next, we need to get the values of trigonometric functions as a function of x. Consider again the given inverse trigonometric functions in order to get the trigonometric functions as a function of x as follows:

Let 

Draw and label a right triangle in order to get the other trigonometric functions as a function of x

Photo by Math Principles in Everyday Life

Let

Draw and label a right triangle in order to get the other trigonometric functions as a function of x

Photo by Math Principles in Everyday Life

Finally, consider again the function

Substitute the values of trigonometric functions to the above equation, we have

Rationalize the denominator in order to eliminate the radical sign

Therefore,

Proving - Parallel Lines, 2

Category: Plane Geometry

"Published in Newark, California, USA"

Given: In ∆ABC, D is the midpoint of AC; DF bisects ∠BDC and ED bisects ∠ADB.

Prove: EF ║ AC

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure and we will label further as follows

Photo by Math Principles in Everyday Life

Proof:

1. Statement: D is the midpoint of line AC.

    Reason: Given item

2. Statement: ED bisects ∠ADB and DF bisects BDC.

    Reason: Given items

3. Statement: ∠ADE ≅ ∠EDB and ∠CDF ≅ ∠FDB.

    Reason: Two bisected angles are congruent to each other.

4. Statement: ∠ADE + ∠EDB + ∠BDF + ∠FDC = 180°

    Reason: The sum of the angles of a straight line is always equal to 180°.

5. Statement: ∠EDB and ∠BDF are complementary angles.

    Reason: From the equation of Statement #4,

               ∠ADE + ∠EDB + ∠BDF + ∠FDC = 180°

    but ∠ADE ≅ ∠EDB and ∠CDF ≅ ∠FDB

    then the above equation becomes

                         2∠EDB + 2∠BDF = 180°

                         2(∠EDB + ∠BDF) = 180°

                             ∠EDB + ∠BDF = 90°
   
6. Statement: ∆EDF is a right triangle.

    Reason: ∠EDF is equal to 90°. ∠EDB and ∠BDF are complementary angles.

7. Statement: EF is the hypotenuse of a rt∆EDF.

    Reason: EF is the opposite side of a right angle of a right triangle. EF is the longest side of a right triangle.

8. Statement: EO ≅ OF

    Reason: If BD is a line bisector of line AC, D is the midpoint of line AC, and D is also a vertex of a rt∆EDF which is a right angle, then line EF which is the hypotenuse of a right triangle will be bisected into EO and OF.

9. Statement: ∆EDO and ∆DOF are isosceles triangles.

    Reason: If BD is a line bisector that passes through the midpoint of the hypotenuse and a vertex of a right triangle which is the opposite angle of the hypotenuse, then BD will bisect a right triangle into two isosceles triangles.

10. Statement: EO ≅ OD and OD ≅ OF.

      Reason: The two sides of an isosceles triangle are congruent.

11. Statement: ∠OED ≅ ∠ODE and ∠ODF ≅ ∠OFD.

      Reason: The two base angles of an isosceles triangles are congruent.

12. Statement: If ∠OED ≅ ∠ODE and ∠ODE ≅ ∠ADE, then ∠OED ≅ ∠ADE.

      Reason: Transitive property of congruence.

13. Statement: If ∠OFD ≅ ∠ODF and ∠ODF ≅ ∠CDF, then ∠OFD ≅ ∠CDF.

      Reason: Transitive property of congruence.

14. Statement: EF ║ AC

      Reason: If the alternating interior angles of a transversal line are congruent, then the two lines that are adjacent to the interior angles are parallel to each other. In the first case, line ED is the transversal line and ∠FED and ∠ADE are the alternating interior angles while in the second case, line DF is the transversal line and ∠EFD and ∠FDC are the alternating interior angles.

Proving - Parallel Lines

Category: Plane Geometry

"Published in Newark, California, USA"

Given: AB ║CD; AF ║EC

Prove: ∠BAF ≅ ∠ECD

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure and we will label further as follows

Photo by Math Principles in Everyday Life

Proof:

1. Statement: AB ║CD; AF ║EC

    Reason: Given items

2. Statement: Draw a line from point A to point C.

    Reason: Line AC will be used as a transversal line.

3. Statement: ∠FAC ≅ ∠ACE

    Reason: If a transversal line AC passes through the parallel lines AF and CE, then their alternating interior angles are congruent.

4. Statement: ∠BAC ≅ ∠ACD

    Reason: If a transversal line AC passes through the parallel lines AB and CD, then their alternating interior angles are congruent.

5. Statement: m∠BAF = m∠BAC - m∠FAC
                      m∠ECD = m∠ACD - m∠ACE

    Reason: Subtraction properties of angles

6. Statement: ∠BAF ≅ ∠ECD

    Reasons: If ∠BAC ≅ ∠ACD and ∠FAC ≅ ∠ACE, then it follows that ∠BAF ≅ ∠ECD.