Trigonometry, which involves the utilization of trigonometric ratios such as sine, cosine, and tangent, serves the purpose of calculating the angles or lengths within a triangle.

In a right angle triangle, the following terms are defined:

• Hypotenuse / 'hyp': the side positioned opposite the 90º angle.
• Opposite / 'opp': the side that is opposite to the angle under consideration.
• Adjacent / 'adj': the side adjacent to the angle being considered.

The identification of the 'opposite' and 'adjacent' sides relies on the specific placement of the angle being analyzed.

Trigonometric Ratios

Within the provided triangle ∆ABC, where AB = c, BC = a, angle C = θ, and angle B = 90⁰, we encounter the ensuing proportions:

• The sine of angle θ represents the quotient of the side 'opposite to the angle' to the 'hypotenuse'.
• Put differently, Sinθ = opp/hyp or AB/AC, which is tantamount to c/b.
• The cosine of angle θ designates the ratio of the side 'adjacent to the angle' to the 'hypotenuse'.
• In other words, Cosθ = adj/hyp or BC/AC, which is synonymous with a/b.
• The tangent of angle θ exemplifies the relation between the side 'opposite to θ' and the side 'adjacent to θ'.
• To clarify, Tanθ = opp/adj or AB/BC, which is analogous to c/a.
• N.B. tanθ = sinθ/cosθ.

Finding missing lengths

Given: ∆ABC, where AB = 6cm, angle B = 90º, angle c = 50º

Find CB & AC

For CB, (use tanθ ratio) tanθ = opp/adj

tan 50º = CB/6

⇒CB = 6 tan50º

CB = 7.15 cm (3 sf)

For AC, we can use:

1. sinθ ratio
2. cosθ ratio
3. Pythagoras' Theorem

Finding missing angles

Given ∆ABC, where AB = 7.4 cm, BC = 7.5cm , AC = 10.5cm, angle B = 90º

To find θ,

tanθ = opp/adj

∴tanθ = 7.5/7.4

θ = tan^-1 (7.5/7.4)

θ = 45.4º

p.s. angle C = 180º - (angle B + θ)

= 180º - (90º + 45.5º)

= 44.6º

Angles of Elevation & Depression

Recall: Alternate angles

• Since both horizontal lines are parallel, it can be seen that θ = alpha (angle of depression = angle of elevation)

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Christopher Seebaran (pc)