If cos2\(\theta\) + \(\frac{1}{8}\) = sin2\(\theta\), find tan\(\theta\)...
Question
If cos2\(\theta\) + \(\frac{1}{8}\) = sin2\(\theta\), find tan\(\theta\)Options
A) 3
B) \(\frac{3\sqrt{7}}{7}\)
C) 3\(\sqrt{7}\)
D) \(\sqrt{7}\)
The correct answer is B.
Explanation:
cos2\(\theta\) + \(\frac{1}{8}\) = sin2\(\theta\)..........(i)
from trigometric ratios for an acute angle, where cos\(\theta\) + sin2\(\theta\) = 1 - cos\(\theta\) ........(ii)
Substitute for equation (i) in (i) = cos2\(\theta\) + \(\frac{1}{8}\) = 1 - cos2\(\theta\)
= cos2\(\theta\) + cos2\(\theta\) = 1 - \(\frac{1}{8}\)
2 cos2\(\theta\) = \(\frac{7}{8}\)
cos2\(\theta\) = \(\frac{7}{2 \times 3}\)
\(\frac{7}{16}\) = cos\(\theta\)
\(\sqrt{\frac{7}{16}}\) = \(\frac{\sqrt{7}}{4}\)
but cos \(\theta\) = \(\frac{\text{adj}}{\text{hyp}}\)
opp2 = hyp2 - adj2
opp2 = 42 (\(\sqrt{7}\))2
= 16 - 7
opp = \(\sqrt{9}\) = 3
than \(\theta\) = \(\frac{\text{opp}}{\text{hyp}}\)
= \(\frac{3}{\sqrt{7}}\)
\(\frac{3}{\sqrt{7}}\) x \(\frac{7}{\sqrt{7}}\) = \(\frac{3\sqrt{7}}{7}\)
from trigometric ratios for an acute angle, where cos\(\theta\) + sin2\(\theta\) = 1 - cos\(\theta\) ........(ii)
Substitute for equation (i) in (i) = cos2\(\theta\) + \(\frac{1}{8}\) = 1 - cos2\(\theta\)
= cos2\(\theta\) + cos2\(\theta\) = 1 - \(\frac{1}{8}\)
2 cos2\(\theta\) = \(\frac{7}{8}\)
cos2\(\theta\) = \(\frac{7}{2 \times 3}\)
\(\frac{7}{16}\) = cos\(\theta\)
\(\sqrt{\frac{7}{16}}\) = \(\frac{\sqrt{7}}{4}\)
but cos \(\theta\) = \(\frac{\text{adj}}{\text{hyp}}\)
opp2 = hyp2 - adj2
opp2 = 42 (\(\sqrt{7}\))2
= 16 - 7
opp = \(\sqrt{9}\) = 3
than \(\theta\) = \(\frac{\text{opp}}{\text{hyp}}\)
= \(\frac{3}{\sqrt{7}}\)
\(\frac{3}{\sqrt{7}}\) x \(\frac{7}{\sqrt{7}}\) = \(\frac{3\sqrt{7}}{7}\)
More Past Questions:
Dicussion (1)
Other Subjects
- English Language
- Biology
- Government
- Physics
- Chemistry
- Economics
- Christian Religious Knowledge
- Commerce
- Geography
- Literature In English
- Accounts
- Agricultural Science
- General Paper
- Islamic Religious Knowledge
- History
- Further Mathematics
- Current Affairs
- Civic Education
- Computer Studies
- Yoruba
- Hausa
- Igbo
- French
- Home Economics
- Sweet Sixteen
- Fine Arts
cos2\(\theta\) + \(\frac{1}{8}\) = sin2\(\theta\)..........(i)
from trigometric ratios for an acute angle, where cos\(\theta\) + sin2\(\theta\) = 1 - cos\(\theta\) ........(ii)
Substitute for equation (i) in (i) = cos2\(\theta\) + \(\frac{1}{8}\) = 1 - cos2\(\theta\)
= cos2\(\theta\) + cos2\(\theta\) = 1 - \(\frac{1}{8}\)
2 cos2\(\theta\) = \(\frac{7}{8}\)
cos2\(\theta\) = \(\frac{7}{2 \times 3}\)
\(\frac{7}{16}\) = cos\(\theta\)
\(\sqrt{\frac{7}{16}}\) = \(\frac{\sqrt{7}}{4}\)
but cos \(\theta\) = \(\frac{\text{adj}}{\text{hyp}}\)
opp2 = hyp2 - adj2
opp2 = 42 (\(\sqrt{7}\))2
= 16 - 7
opp = \(\sqrt{9}\) = 3
than \(\theta\) = \(\frac{\text{opp}}{\text{hyp}}\)
= \(\frac{3}{\sqrt{7}}\)
\(\frac{3}{\sqrt{7}}\) x \(\frac{7}{\sqrt{7}}\) = \(\frac{3\sqrt{7}}{7}\)