IDNLearn.com: Where curiosity meets clarity and questions find their answers. Join our knowledgeable community to find the answers you need for any topic or issue.
Sagot :
To determine the solutions to the equation [tex]\(8 \cos^2 \theta - 3 \cos \theta = 0\)[/tex] in the interval [tex]\(0^\circ \leq \theta \leq 180^\circ\)[/tex], follow these steps:
1. Rewrite the equation in a simpler form:
Let [tex]\( x = \cos \theta \)[/tex]. The equation becomes:
[tex]\[ 8x^2 - 3x = 0 \][/tex]
2. Factor the quadratic equation:
[tex]\[ x(8x - 3) = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad 8x - 3 = 0 \][/tex]
Solving [tex]\( 8x - 3 = 0 \)[/tex]:
[tex]\[ 8x = 3 \implies x = \frac{3}{8} \][/tex]
4. Convert back to [tex]\( \theta \)[/tex]:
Recall [tex]\( x = \cos \theta \)[/tex].
- For [tex]\( \cos \theta = 0 \)[/tex]:
[tex]\[ \theta = 90^\circ \][/tex]
- For [tex]\( \cos \theta = \frac{3}{8} \)[/tex]:
[tex]\[ \theta \approx 68.0^\circ \][/tex]
Therefore, the possible values of [tex]\( \theta \)[/tex] in the interval [tex]\([0^\circ, 180^\circ]\)[/tex] are [tex]\( 90.0^\circ \)[/tex] and [tex]\( 68.0^\circ \)[/tex].
Checking the given choices:
- [tex]\( 0.0^\circ \)[/tex]: Not a solution.
- [tex]\( 22.0^\circ \)[/tex]: Not a solution.
- [tex]\( 52.0^\circ \)[/tex]: Not a solution.
- [tex]\( 68.0^\circ \)[/tex]: Solution.
- [tex]\( 90.0^\circ \)[/tex]: Solution.
Thus, the solutions are [tex]\( 68.0^\circ \)[/tex] and [tex]\( 90.0^\circ \)[/tex].
1. Rewrite the equation in a simpler form:
Let [tex]\( x = \cos \theta \)[/tex]. The equation becomes:
[tex]\[ 8x^2 - 3x = 0 \][/tex]
2. Factor the quadratic equation:
[tex]\[ x(8x - 3) = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad 8x - 3 = 0 \][/tex]
Solving [tex]\( 8x - 3 = 0 \)[/tex]:
[tex]\[ 8x = 3 \implies x = \frac{3}{8} \][/tex]
4. Convert back to [tex]\( \theta \)[/tex]:
Recall [tex]\( x = \cos \theta \)[/tex].
- For [tex]\( \cos \theta = 0 \)[/tex]:
[tex]\[ \theta = 90^\circ \][/tex]
- For [tex]\( \cos \theta = \frac{3}{8} \)[/tex]:
[tex]\[ \theta \approx 68.0^\circ \][/tex]
Therefore, the possible values of [tex]\( \theta \)[/tex] in the interval [tex]\([0^\circ, 180^\circ]\)[/tex] are [tex]\( 90.0^\circ \)[/tex] and [tex]\( 68.0^\circ \)[/tex].
Checking the given choices:
- [tex]\( 0.0^\circ \)[/tex]: Not a solution.
- [tex]\( 22.0^\circ \)[/tex]: Not a solution.
- [tex]\( 52.0^\circ \)[/tex]: Not a solution.
- [tex]\( 68.0^\circ \)[/tex]: Solution.
- [tex]\( 90.0^\circ \)[/tex]: Solution.
Thus, the solutions are [tex]\( 68.0^\circ \)[/tex] and [tex]\( 90.0^\circ \)[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is committed to your satisfaction. Thank you for visiting, and see you next time for more helpful answers.