Question : If $21 \tan \theta=20$, then $(1+\sin \theta+\cos \theta):(1-\sin \theta+\cos \theta)=$?
Option 1: 5 : 2
Option 2: 3 : 1
Option 3: 7 : 3
Option 4: 2 : 1
Correct Answer: 7 : 3
Solution :
Given: $21 \tan \theta=20$
⇒ $\tan \theta = \frac{20}{21}=\frac{\text{Perpendicular}}{\text{Base}}$
Let the perpendicular and the base be 20 units and 21 units respectively.
Use the formulas,
$\tan \theta=\frac{\text{Perpendicular}}{\text{Base}}$,
$\sin \theta=\frac{\text{Perpendicular}}{\text{Hypotenuse}}$,
$\cos \theta=\frac{\text{Base}}{\text{Hypotenuse}}$,
Pythagoras's theorem: $\text{Hypotenuse}^2=\text{Base}^2+\text{Perpendicular}^2$
⇒ $\text{Hypotenuse}^2={21}^2+{20}^2$
⇒ $\text{Hypotenuse}^2=441+400$
⇒ $\text{Hypotenuse}^2=841$
⇒ $\text{Hypotenuse}=\sqrt{841}$
⇒ $\text{Hypotenuse}=29$
The value of $(1+\sin \theta+\cos \theta):(1-\sin \theta+\cos \theta)$ is given as,
$=(1+\frac{20}{29}+\frac{21}{29}):(1-\frac{20}{29}+\frac{21}{29})$
$=\frac{29+20+21}{29}:\frac{29–20+21}{29}$
$=\frac{70}{29}:\frac{30}{29}=7:3$
Hence, the correct answer is 7 : 3.
Related Questions
Question : If $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$, then $(\cos \theta+\sin \theta)$ is:
Question : If $\cos ^2 \theta-\sin ^2 \theta=\tan ^2 \phi$, then which of the following is true?
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