Some Trigonometric Ratios

Consider a right-angled triangle ABC below.

The angle θ can be expressed in terms of cosine ratio or sine ratio. The hypotenuse of the triangle has dimension r.

Adjacent side to angle θ is the line AB and the opposite side to the angle θ is line BC.

expressing angle θ in terms of cosine ratio and sin ratio:

by use of Pythagoras theorem:

(AB)² + (BC)² = r²

(rcosθ)² + (rcosθ)² = r² and hence

r² cos² θ + r² sin² θ = r²

then dividing everywhere by r² , then we get

cos² θ + sin² θ =1 which is a trigonometric identity which holds true for all values of θ.

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If tan θ = a, show that:

we factor out cosθ to get:

but sin² θ + cos² θ =1

hence:

but tanθ = a, so 1/tanθ = 1/a

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