Verifying Trigonometry Identities
Verifying Trigonometry Identities - How it Works - Video
Trigonometry Fundamental and Pythagorean Identities
Trigonometry Fundamental and Pythagorean Identities:
Here we have the building blocks for verifying identities.
Example 1A
Example 1A:
Here we have to verify the identity sec θ - cos θ = sin θ * tan θ. We can work on the left side until we match the right side or we can work on the right side until we match the left side. After about 5 steps or so, if you aren't close to matching the other side, it might be wise to switch sides.
For this example we choose to the left hand side of our equation to manipulate.
Line 0 - We chose the left hand side.
Line 1 - We used one of the Fundamental Identities.
Line 2 - We multiplied by 1 or cos θ / cos θ.
Line 3 - We multiplied the cos θs.
Line 4 - We combined the fraction.
Line 5 - We use one of the Pythagorean Identities.
Line 6 - We wrote the last step again.
Line 7 - We separated the sin2 θ.
Line 8 - We separated the sin θs.
Line 9 - We use one of the Fundamental Identities.
=> sec θ - cos θ
=> 1 / cos θ - cos θ
=> 1 / cos θ - cos θ * [ cos θ / cos θ]
=>1 / cos θ - cos2 θ / cos θ
=> [1 - cos2 θ ] / cos θ
=> sin2 θ / cos θ
=> sin2 θ / cos θ
=> [sin θ * sin θ] / cos θ
=> sin θ *[ sin θ / cos θ]
=> sin θ * tan θ
Definitions:
Here we have the definition of a geometric sequence. The formula is ak+1= ak * r, which states that the next term is equal to the previous times a ratio.
The formula for Nth term is an= a1 * rn-1, which states the Nth term is equal to the first term times the ratio raised to power of nth term minus one. θ
Example 1B
Example 1B:
Here we have to verify the identity sec θ - cos θ = sin θ * tan θ. We can work on the left side until we match the right side or we can work on the right side until we match the left side. After about 5 steps or so, if you aren't close to matching the other side, it might be wise to switch sides.
For this example we choose to the right hand side of our equation to manipulate.
Line 0 - We chose the right hand side.
Line 1 - We used one of the Fundamental Identities.
Line 2 - We multiplied the sin θs.
Line 3 - We use one of the Pythagorean Identities.
Line 4 - We separate into two fractions.
Line 5 - We used one of the Fundamental Identities.
Line 6 - We divided by cos θ.
=> sin θ * tan θ
=> sin θ * [sin θ / cos θ]
=> sin2 θ / cos θ
=> [1 - cos2 θ] / cos θ
=>1 / cos θ - cos2 θ / cos θ
=> sec θ - cos2 θ / cos θ
=> sec θ - cos θ
Example 2A
Example 2A:
Here we have to verify the identity [sec2 (2α) - 1] / sec2 (2α) = sin (2α). We can work on the left side until we match the right side or we can work on the right side until we match the left side. After about 5 steps or so, if you aren't close to matching the other side, it might be wise to switch sides.
For this example we choose to the left hand side of our equation to manipulate.
Line 0 - We chose the left hand side.
Line 1 - We used one of the Pythagorean Identities.
Line 2 - We used one of the Fundamental Identities.
Line 3 - We flipped the second fraction.
Line 4 - We used one of the Fundamental Identities.
Line 5 - We multiplied the cos2 (2α)s.
=> [sec2 (2α) - 1] / sec2 (2α)
=> tan2 (2α) / sec2 (2α)
=> tan2 (2α) / [1 / cos2 (2α)]
=> tan2 (2α) * cos2 (2α)
=> [sin2 (2α) / cos2 (2α)] * cos2 (2α)
=> sin2 (2α)
Example 2B
Example 2B:
Here we have to verify the identity [sec2 (2α) - 1] / sec2 (2α) = sin (2α). We can work on the left side until we match the right side or we can work on the right side until we match the left side. After about 5 steps or so, if you aren't close to matching the other side, it might be wise to switch sides.
For this example we choose to the right hand side of our equation to manipulate.
Line 0 - We chose the right hand side.
Line 1 - We used one of the Pythagorean Identities.
Line 2 - We used one of the Fundamental Identities.
Line 3 - We multiplied by 1 or sec2 (2α) / sec2 (2α)
Line 4 - We multiplied.
Line 5 - We combined fractions.
=> sin (2α)
=> 1 - cos2 (2α)
=> 1 - 1 / sec2 (2α)
=> [sec2 (2α) / sec2 (2α)] *1 - [1 / sec2 (2α)]
=> sec2 (2α) / sec2 (2α)] - [1 / sec2 (2α)]
=> [sec2 (2α) - 1] / sec2 (2α)