If tan θ = 2/3, then sin θ = ?
Trigonometry is a very important branch of Mathematics. The famous Greek Astronomer and mathematician Hipparchus was the inventor of this special branch of mathematics. The name ‘Trigonometry’ implies that the method of measurement relative to triangles is the subject of trigonometry (note that ‘trigonon’ means triangle and ‘metron’ means measurement method). To put it clearly, the special branch of mathematics that deals with three sides of a triangle, its three angles, its area, and their interrelationships is called Trigonometry.
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In this right-angled triangle, let the reference angle between base and hypotenuse is θ (theta). On the longest side, the hypotenuse is directly opposite to the right angle, which is also the largest angle among the three angles.
In a right-angle triangle, sinθ = perpendicular / hypotenuse; cosθ = base / hypotenuse ; tanθ = perpendicular / base
Example, tan A = (perpendicular / hypotenuse) / (base / hypotenuse)
= sin A / cos A , where A is any angle without 90° in right angle triangle.
Some extra trigonometric rules,
cotθ = 1 / tanθ = base / perpendicular
secθ = 1 / cosθ = hypotenuse / base
cosecθ = 1 / sinθ = hypotenuse / perpendicular, where θ is any angle without 90°
The values of trigonometric expectant of any angle are either positive or negative. From the definitions of trigonometric expectant, in this picture,
sin A = BC/AB and cos A = CA/AB
Now, AB is the hypotenuse of the right-angled triangle ABC. Thus, the value of AB is always greater than BC & CA. So, the value of sin A and cos A always be less than or equal to 1.
then, cosec A = AB/BC and sec A = AB/CA
So, the value of cosec A and sec A always be greater than or equal to 1
Also, tan A = BC/CA and cot A = CA/BC
Clearly, the value of BC can be either larger or smaller than CA. So, In a right-angled triangle, the hypotenuse is the largest side among them. θ is a positive angle, then:
1. The value of sinθ, cosθ cannot be greater than 1.
2. The value of cosecθ, secθ cannot be less than 1.
3. Tanθ, cotθ have any value.
Angle ratios (θ) 0° 30° 45° 60° 90° sin (sine) 0 1/2 1/√2 √3/2 1 cos (cosine) 1 √3/2 1/√2 1/2 0 tan (tangent) 0 1/√3 1 √3 ∞ cot (cotangent) ∞ √3 1 1/√3 0 sec (secant) 1 2/√3 √2 2 ∞ cosec (cosecant) ∞ 2 √2 2/√3 1
Some Important formulae
- sin2A + cos2A = 1
- sec2A – tan2A = 1 and,
- cosec2A – cot2A = 1
If tan θ = 2/3, then sin θ =?
Now, If tan A = 2/3, cot A = 3/2
Then, cosec2A = 1 + cot2A [from 3rd rule]
= 1 + (3/2)2
= 1 + 9/4
So, cosec A = √(13/4) = √13/2
So, sin A = 1 / cosec A = 2/√13
sin A = 2 / √13.
Question 1: If tanθ = -1/√5, then cosθ = ?
tanθ = -1/√5
Then, sec2θ = 1 + tan2θ
= 1 + (-1/√5)2
= 1 + (1/5)
⇒ sec2θ = 6/5
⇒ secθ = √(6/5)
So, cosθ = 1 / secθ = √(5/6)
cosθ = √(5/6)
Question 2: If cosecφ = 2/√3, then φ = ?
cosecφ = 2/√3
⇒ φ = cosec-1(2/√3) = 60°
φ = 60°
Question 3: If tanα = 5/12 and sinα is -ve(negative), then cosα = ?
tanα = 5/12
Then, sec2α = 1 +tan2α
= 1 + (5/12)2
= 1 + (25/144)
⇒ sec2α = 169/144
⇒ secα = √(169/144) = ±(13/12)
In this question, sinα = -ve (negative) (given) and tanα is positive, so cosα is obviously negative
Then, cosα = -12/13
cosα = -12/13