if cos theta =5/3 then find the value of cos theta + tan theta
Answers
Step-by-step explanation:
Hello ,
This problem can be solved by using identities
\csc( \theta) = \frac{5}{3}csc(θ)=
3
5
{ \csc( \theta) }^{2} = { \cot( \theta) }^{2} + 1csc(θ)
2
=cot(θ)
2
+1
{( \frac{5}{3}) }^{2} = { \cot( \theta) }^{2} + 1(
3
5
)
2
=cot(θ)
2
+1
{ \cot( \theta) }^{2} = \frac{25}{9} - 1cot(θ)
2
=
9
25
−1
\cot( \theta) = \sqrt{ \frac{16}{9} }cot(θ)=
9
16
\cot( \theta) = \frac{4}{3}cot(θ)=
3
4
\cot( \theta) = \frac{1}{ \tan( \theta) }cot(θ)=
tan(θ)
1
\tan( \theta) = \frac{3}{4}tan(θ)=
4
3
1 + tan^2 theta = sec^2 theta
sec^2 theta = 1 + 9/16
= 25/16
sec theta = √25/16
= 5/4
cos theta = 4/5
\begin{gathered} \cos( \theta) + \tan( \theta) = \frac{4}{5} \: + \frac{3}{4} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{16 + 15}{20} \end{gathered}
cos(θ)+tan(θ)=
5
4
+
4
3
=
20
16+15
(16 + 15)/20 = 31/20