Answer:
GIVEN :-
Coordinates of points are :-
- (-5 , 12)
- (2 , 8)
- (3 , -6)
TO FIND :-
- All the trigonometric values of given points
FACTS TO KNOW BEFORE SOLVING :-
It's important to know that :-
- In 1st quadrant (0° to 90°) , all the trigonometric values are positive .
- In 2nd quadrant (90° to 180°) , except sin & cosec , rest all trigonometric values are negative.
- In 3rd quadrant (180° to 270°) , except tan & cot , rest all trigonometric values are negative.
- In 4th quadrant (270° to 360°) , except cos & sec , rest all all trigonometric values are negative.
SOLUTION :-
Q1)
- Plot (-5,12) on the cartesian plane and name it 'A'
- Drop a perpendicular to x-axis from 'A' and name the point 'B' where the perpendicular meets x-axis.
- Join the point A with the origin 'O'.
You'll notice a right-angled ΔABO formed (∠B = 90°) in the 2nd quadrant of the plane whose :-
- length of perpendicular of triangle (AB) = 12 units
- length of base of triangle (OB) = 5 units
- length of hypotenuse (OA) = 13 units
Let the angle between OA & positive x-axis be θ.
⇒ ∠AOB = 180 - θ
So ,
- [tex]\sin (AOB) = \sin(180 - \theta) = \sin \theta = \frac{12}{13}[/tex]
- [tex]\cos(AOB) = \cos (180 - \theta) = -\cos \theta = -\frac{5}{13}[/tex]
- [tex]\tan(AOB) = \tan(180 - \theta) = -tan \theta = -\frac{12}{5}[/tex]
- [tex]\csc(AOB) = \csc(180 - \theta) = \csc \theta = \frac{1}{\sin \theta} = \frac{13}{12}[/tex]
- [tex]\sec(AOB) = \sec(180 - \theta) = -\sec \theta = -\frac{1}{\cos \theta} = -\frac{13}{5}[/tex]
- [tex]\cot(AOB) = \cot(180 - \theta) = -\cot \theta = -\frac{1}{\tan \theta} = -\frac{5}{12}[/tex]
Q2)
- Plot (2,8) on the cartesian plane and name it 'A'
- Drop a perpendicular to x-axis from 'A' and name the point 'B' where the perpendicular meets x-axis.
- Join the point A with the origin 'O'.
You'll notice a right-angled ΔABO formed (∠B = 90°) in the 1st quadrant of the plane whose :-
- length of perpendicular of triangle (AB) = 8 units
- length of base of triangle (OB) = 2 units
- length of hypotenuse (OA) = 2√17 units
Let the angle between OA & positive x-axis be θ.
⇒ ∠AOB = θ
So ,
- [tex]\sin(AOB) = \sin \theta = \frac{8}{2\sqrt{17} } = \frac{4}{\sqrt{17} }[/tex]
- [tex]\cos(AOB) = \cos \theta = \frac{2}{2\sqrt{17}} = \frac{1}{\sqrt{17}}[/tex]
- [tex]\tan(AOB) = \tan \theta = \frac{8}{2} = 4[/tex]
- [tex]\csc(AOB) = \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{17}}{4}[/tex]
- [tex]\sec(AOB) = \sec \theta = \frac{1}{\cos \theta} = \sqrt{17}[/tex]
- [tex]\cot (AOB) = \cot \theta = \frac{1}{\tan \theta} = \frac{1}{4}[/tex]
Q3)
- Plot (3,-6) on the cartesian plane and name it 'A'
- Drop a perpendicular to x-axis from 'A' and name the point 'B' where the perpendicular meets x-axis.
- Join the point A with the origin 'O'.
You'll notice a right-angled ΔABO formed (∠B = 90°) in the 4th quadrant of the plane whose :-
- length of perpendicular of triangle (AB) = 6 units
- length of base of triangle (OB) = 3 units
- length of hypotenuse (OA) = 3√5 units
Let the angle between OA & positive x-axis be θ . [Assume it in counterclockwise direction].
⇒ ∠AOB = 360 - θ
So ,
- [tex]\sin(AOB) = \sin(360 -\theta) = -\sin \theta = -\frac{6}{3\sqrt{5} } = -\frac{2}{\sqrt{5} }[/tex]
- [tex]\cos(AOB) = \cos(360 - \theta) = \cos \theta = \frac{3}{3\sqrt{5} } = \frac{1}{\sqrt{5} }[/tex]
- [tex]\tan(AOB) = \tan(360 - \theta) = -tan \theta = -\frac{6}{3} = -2[/tex]
- [tex]\csc(AOB) = \csc(360 - \theta) = -\csc \theta = -\frac{1}{\sin \theta} = -\frac{\sqrt{5} }{2}[/tex]
- [tex]\sec(AOB) =\sec (360 - \theta) = \sec \theta = \frac{1}{\cos \theta} = \sqrt{5}[/tex]
- [tex]\cot(AOB) = \cot(360 - \theta) = -\cot \theta = -\frac{1}{\tan \theta} = -\frac{1}{2}[/tex]
