Lesson Example Discussion Quiz: Class Homework |
Step-4 |
Title: Inverse Trigonometric Functions |
Grade: 10-a Lesson: S3-L8 |
Explanation: Hello Students, time to practice and review the steps for the problem. |
Lesson Steps
| Step | Type | Explanation | Answer |
|---|---|---|---|
1 |
Problem |
Simplify the following inverse trigonometry function: \$cos^-1(x) + cos^-1(y) + cos^-1(z) = (pi)\$. |
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2 |
Step |
The given function is |
\$cos^-1(x) + cos^-1(y) + cos^-1(z) = (pi)\$ |
3 |
Formula: |
The given function is written as and simplify it |
\$cos^-1(x) + cos^-1(y) = (pi) - cos^-1(z)\$ \$cos^-1(x) + cos^-1(y) = cos^-1(- z)\$ Since, \$cos^-1(- θ) = (pi) - cos^-1 (θ)\$ |
4 |
Clue |
The sum of inverse cosines formula is |
\$cos^-1(x) + cos^-1(y) = cos^-1 (xy - \sqrt(1 - x^2) . \sqrt(1 - y^2))\$ |
5 |
Hint |
Substitute the formula into the simplified function and then simplify it |
\$cos^-1 (xy - \sqrt(1 - x^2) . \sqrt(1 - y^2)) = cos^-1(- z)\$ \$cos^-1(x) + cos^-1(y) = cos^-1(- z)\$ Since, \$cos^-1(- θ) = (pi) - cos^-(θ)\$ |
6 |
Clue |
The sum of inverse cosines formula is |
\$cos^-1(x) + cos^-1(y) = cos^-1 (xy - \sqrt(1 - x^2) . \sqrt(1 - y^2))\$ |
7 |
Hint |
Substitute the formula into the simplified function and then simplify it |
\$cos^-1 (xy - \sqrt(1 - x^2) . \sqrt(1 - y^2)) = cos^-1(- z)\$ \$xy - \sqrt(1 - x^2) . \sqrt(1 - y^2) = - z\$ \$xy + z = \sqrt(1 - x^2) . \sqrt(1 - y^2)\$ |
8 |
Step |
Now squaring on both sides then simplify |
\$(xy + z)^2 = (\sqrt(1 - x^2))^2 .( \sqrt(1 - y^2))^2\$ \$x^2 + y^2 + z^2 + 2xyz = 1\$ |
9 |
Step |
Therefore, the inverse trigonometry function: \$cos^-1(x) + cos^-1(y) + cos^-1(z) = (pi)\$ is \$x^2 + y^2 + z^2 + 2xyz = 1\$. |
|
10 |
Choice.A |
Wrong: Because it incorrectly suggests a different relationship that does not satisfy the equation derived from the inverse trigonometric function identities |
\$x^2 + y^2 - z^2 + 2xyz = 1\$ |
11 |
Choice.B |
Option B is not correct because it does not match the derived and validated identity for the sum of the inverse cosines equating to π |
\$x^2 - y^2 + z^2 - 2xyz = 1\$ |
12 |
Choice.C |
It does not satisfy the equation formed by the inverse cosine identities, and it does not correctly simplify to match the original equation involving the sum of inverse cosines |
\$x^2 - y^2 + z^2 - 2xyz = - 1\$ |
13 |
Choice.D |
It satisfies the equation derived from the inverse cosine identities and correctly simplifies to match the original equation involving the sum of inverse cosines |
\$x^2 + y^2 + z^2 + 2xyz = 1\$ |
14 |
Answer |
Option |
D |
15 |
Sumup |
Can you summarize what you’ve understood in the above steps? |
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