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Introduction |
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An equation containing the trigonometrical functions of an unknown quantity is termed as a trigonometrical equation, which holds for some values (and not for all values) of the quantities involved. |
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Co-terminal Angles |
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The angles (2p + A), (4p + A)......have the same initial and terminal arm as the angle A and so these angles are called co-terminal angles. For all such angles, the value of any trigonometric ratio is the same. Thus, all co-terminal angles are trigonometrically equivalent. |
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Theorem 1 |
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General solution of sin q = k. |
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Theorem 2 |
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General solution of cos q = k. |
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Theorem 3 |
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General solution of tan q = k. |
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Theorem 4 |
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General solution of acosq + bsinq = c. |
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Sine Formula (Sine Rule) |
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Prove that in any triangle, the sides of a triangle are proportional to the sines of the opposite angles. |
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Cosine Formula (Cosine Rule) |
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If ABC is a triangle with sides a = BC, b = CA, c = AB, then prove that |
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a2 = b2 + c2 - 2 bc cos A,
b2 = c2 + a2 - 2 ca cos B,
c2 = a2 + b2 - 2 ab cos C. |
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Projection Formula |
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In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove
that
a = b cos C + c cos B, b = c cos A + a cos C, c
= a cos B + b cos A |
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Napier's Analogy |
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In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove that |
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Area of a triangle |
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In any triangle ABC, prove that the area of a triangle ABC is given by |
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Hero's Formula |
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Half-Angle Formulae |
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If ABC is a triangle with sides a = BC, b = CA, c = AB, then prove that |
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 where 2s = a + b + c. |
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Solution of triangles |
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From geometry, we know that when any three elements are given of which necessarily a side is given, the triangle is completely determined i.e, remaining three elements can be determined. The process of determining the unknown elements knowing the known elements is known as the solution of a triangle. |
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Heights and Distances |
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One of the main use of trigonometry is to find the distances between points, or the heights of objects, without actually measuring these distances or these heights. |
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Inverse Trigonometric Functions |
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For x Î [-1,
1], if q is an angle whose sine is x, then we say
that sine inverse x is q and write sin-1x = q. |
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Summary |
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1. In the principle value branches, the following formulae holds: |
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sin-1 (sin x) = x, cos-1 (cos x) = x, tan-1 (tan x) = x, cos-1 (cot x) = x, sec-1 (secx) = x, cosec-1 (cosecx) = x |
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2. If a is some constant angle, then |
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- sin q = sin a
Þ q = np + (-1)n
a, n Î Z |
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- cos q = cos a
Þ q = 2np ± a, n Î
Z |
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- tan q = tan a
Þ q = np + a, n
Î Z |
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