Trigonometry in Right Angled Triangles

( 1,1 , √ 2 )and ( 1 , √ 3 , 2 )are Pythagorean triples, so triangles with these sides form right angled
triangles as shown in figure above.
Using the SINE law for triangle ( 1,1 , √ 2 ):
1
sin ( c ) = 1
sin ( b ) = √ 2
sin ( 90 )
Implies,
∠ b=∠ c
but,
In a triangle abc :∠ a+∠ b+∠ c=180 °
therefore,
90 ° +∠ b+∠ b=180 ° ⇒ ∠ b=∠c=45 °
Using the SINE law for triangle ( 1 , √ 3 , 2 ):
1
sin ( r ) = 2
sin ( 90 ) = √ 3
sin ( p )
therefore,
sin ( r ) =1 ·sin ( 90 )
2 ⇒sin ( r )= 1
2 ⇒ r =sin−1
( 1
2 )=30°
∠ p+ ∠q +∠r =180 °
⇒ 90° +∠ q+ 30°=180° ⇒∠ q=60 °
a) sin ( 60 ° )

From triangle pqr:
sin ( 60 ° ) =sin ( q ) =side opposite ¿ q ¿
hypotenuse = pr
qr = √ 3
2
b) cos ( 45 ° )
From triangle abc:
cos ( 45 ° ) =cos ( c ) =side adjecent ¿ c ¿
hypotenuse = ac
bc = 1
√ 2
c) tan ( 30 ° )
From triangle pqr:
tan ( 30 ° )=tan ( r )=side opposite ¿ r ¿
side adjecent ¿ r ¿= 1
√3
The Pythagoras rule for right angled triangles states that:
Square of hypotenuse is equal to the sum of squares of remaining two sides.
Hypotenuse is the side opposite to the right angle, which is ‘a’ for first triangle and ‘9’ for the second
triangle.
Using Pythagoras rule for the first triangle:
a2=52 +32
⇒ a= √ 52 +32 = √ 34 ≅ 5.83
Using Pythagoras rule for the second triangle:
92 =z2+82 ⇒ z2 =92−82=17 ⇒ z= √ 17 ≅ 4.12

Consider any right angle triangle abc as shown in the figure above.
Circumcentre of a right angle triangle lies at the midpoint of hypotenuse (=ac).
Point d is the midpoint of hypotenuse ac. As d is the circumcentre, points a, b and c lie on a
circumcircle with radius = da or db or dc. Thus, da = db = dc which is indicated by the labelling in
figure above.
This means that triangle adb is a isosceles triangle with ad = bd, and triangle cdb is also an isosceles
triangle with bd = cd. Thus, with this simple construction, we end up with 2 isosceles triangles from a
right angle triangle.
Equal lengths and equal angles are marked in the figure below:

Trigonometric ratios on a unit circle (radius =1, centred at origin) are shown in figure below.
Form the figure we deduce that for an angle θ, the x- coordinate of the radius at that angle is cos θ,
y- coordinate is sin θ and tanθ is the length of tangent from the point upto the x- axis.
Graph of unit circle showing various angles and coordinates for each of the angles (point of
intersection of radius at the angle and the circle), is shown in the figure below.
a) sin ( 210 ° )