Allied Angles 1c1u4m

Allied Angles

by SBR

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Let 𝜽 be any angle. Then angles, −𝜽, 𝟗𝟎 ± θ , 𝟏𝟖𝟎 ± θ , 𝟐𝟕𝟎 ± θ , 𝟑𝟔𝟎 ± θ are angles allied to 𝜽.

In general, angles of the form 𝒏

𝝅 𝟐

± 𝜽 are called angles allied to 𝜽.

In particular,

• 𝜽 and 𝟗𝟎∘ − 𝜽 are called complementary angles. • 𝜽 and 𝟏𝟖𝟎∘ − 𝜽 are called complementary angles.

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Relation between the trigonometric ratios of −𝜽 and 𝜽 Consider a circle with center 𝑶 and radius 𝒓 units. Let 𝑃 𝑥, 𝑦 be any point on the circle and 𝑄 𝑥, −𝑦 be its image about 𝑥 − 𝑎𝑥𝑖𝑠.

Let the line ing 𝑷 and 𝑸 intersect the 𝒙 − 𝒂𝒙𝒊𝒔 at 𝑴. We have, 𝑂𝑃 = 𝑂𝑄 = 𝑟 and 𝑃𝑀 = 𝑄𝑀 Let ∠𝑋𝑂𝑃 = 𝜃 then ∠𝑋𝑂𝑄 = −𝜃

We have, 𝒔𝒊𝒏 𝜽 =

𝒚 𝒓

, 𝒄𝒐𝒔 𝜽 =

𝒙 𝒓

and 𝒕𝒂𝒏 𝜽 =

𝒚 𝒙

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Relation between the trigonometric ratios of −𝜽 and 𝜽 (contd.) consider , 𝒚′ −𝒚 𝒚 = = − = − 𝒔𝒊𝒏 𝜽 𝒓 𝒓 𝒓 𝒙′ 𝒙 𝒄𝒐𝒔 −𝜽 = = = 𝒄𝒐𝒔 𝜽 𝒓 𝒓 𝒚′ −𝒚 𝒚 𝒕𝒂𝒏 −𝜽 = = = − = − 𝒕𝒂𝒏 𝜽 𝒙′ 𝒙 𝒙 𝒔𝒊𝒏 −𝜽 =

Thus we have,

Taking reciprocals 𝒔𝒊𝒏 −𝜽 = −𝒔𝒊𝒏 𝜽 𝒄𝒐𝒔𝒆𝒄 −𝜽 = −𝒄𝒐𝒔𝒆𝒄 𝜽 𝒄𝒐𝒔 −𝜽 = 𝒄𝒐𝒔 𝜽

𝒔𝒆𝒄 −𝜽 = 𝒔𝒆𝒄 𝜽

𝒕𝒂𝒏 −𝜽 = −𝒕𝒂𝒏 𝜽

𝒄𝒐𝒕 −𝜽 = −𝒄𝒐𝒕 𝜽

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Relation between the trigonometric ratios of (𝟗𝟎° − 𝜽) and 𝜽 Consider a ⊿𝐴𝐵𝐶

Let ∠𝑨𝑩𝑪 = 𝟗𝟎 and ∠𝑪𝑨𝑩 = 𝜽 ∴ ∠ACB = (90 - θ)

Let 𝑨𝑩 = 𝒚, 𝑩𝑪 = 𝒙 and 𝑨𝑪 = 𝒓 We have, 𝒔𝒊𝒏 𝜽 = 𝒙/𝒓 𝒄𝒐𝒔 𝜽 = 𝒚/𝒓 𝒕𝒂𝒏 𝜽 = 𝒙/𝒚

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Relation between the trigonometric ratios of (𝟗𝟎° − 𝜽) and 𝜽 Consider 𝟗𝟎 − 𝜽

We have, 𝑠𝑖𝑛 90∘ − 𝜃 =

𝐴𝐵 𝑦 = = 𝑐𝑜𝑠 𝜃 𝐴𝐶 𝑟

𝑐𝑜𝑠 90∘ − 𝜃 =

𝐵𝐶 𝑥 = = 𝑠𝑖𝑛 𝜃 𝐴𝐶 𝑟

𝑡𝑎𝑛 90∘ − 𝜃 =

𝐴𝐵 𝑦 1 = = = 𝑐𝑜𝑡 𝜃 𝐵𝐶 𝑥 𝑡𝑎𝑛 𝜃

Thus we have,

Taking reciprocals

𝒔𝒊𝒏 𝟗𝟎∘ − 𝜽 = 𝒄𝒐𝒔 𝜽 𝒄𝒐𝒔𝒆𝒄 𝟗𝟎∘ − 𝜽 = 𝒔𝒆𝒄 𝜽 𝒄𝒐𝒔 𝟗𝟎∘ − 𝜽 = 𝒔𝒊𝒏 𝜽 𝒔𝒆𝒄 𝟗𝟎∘ − 𝜽 = 𝒄𝒐𝒔𝒆𝒄 𝜽 𝒕𝒂𝒏 𝟗𝟎∘ − 𝜽 = 𝒄𝒐𝒕 𝜽

𝒄𝒐𝒕 𝟗𝟎∘ − 𝜽 = 𝒕𝒂𝒏 𝜽

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Relation between the trigonometric ratios of (𝟗𝟎° + 𝜽) and 𝜽

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