Day 88 – Trigonometric ratios of complements

Introduction In a right triangle, the measure of the right angle is constant, and the other two acute angles are complementary, that is their sum is 90°. Each angle in the pair is complementary to the other. We have already learned how to define trigonometric ratios of acute angles in our earlier lesson. In this lesson, we will define trigonometric ratios of complementary angles based on the trigonometric ratios of the acute angles we had earlier discussed.

Vocabulary 1. Complementary angles A pair of angles whose sum is 90°. 2. Complement One angle is said to be the complement of another angle if the two angles add up to 90°.

Trigonometric ratios of complementary angles In any right triangle, the to acute angles must always sum up to 90°, therefore they are always complementary. Consider right ∆ABC on the next slide. ∠𝐴 and ∠𝐶 are complementary, that is, ∠𝐴+∠𝐶=90° This shows that since ∠𝐶=𝜃 then ∠𝐴=90°−𝜃.

Trigonometric ratios of ∠𝐴=90°−𝜃, which is complementary to ∠𝐵=𝜃 can be defined using the sides of right ∆ABC as follows: 𝜃 90°−𝜃 A B C 𝑏 𝑐 𝑎

sin 90°−𝜃 = 𝐵𝐶 𝐴𝐶 = 𝑎 𝑏 cos 90°−𝜃 = 𝐴𝐵 𝐴𝐶 = 𝑐 𝑏 tan 90°−𝜃 = 𝐵𝐶 𝐴𝐵 = 𝑎 𝑐

Example Right ΔXYZ below is right angled at Y and ∠𝑋 is given as 𝜃 Example Right ΔXYZ below is right angled at Y and ∠𝑋 is given as 𝜃. Use it to answer the questions below. X Y Z 𝜃 𝑦 𝑧 𝑥

(a) Write ∠𝑍 in terms of 𝜃 (a) Write ∠𝑍 in terms of 𝜃. (b) Express the sine, cosine and tangent of ∠𝑍 in terms of the sides 𝑥, 𝑦 and 𝑧.

Solution (a) ∠𝑋 and ∠𝑍 are complements to each other, hence, ∠𝑋+ ∠𝑍=90° 𝜃+ ∠𝑍=90° ∴∠𝑍=90°−𝜃 (b) We use the acronym SOH CAH TOA to identify the sides with reference to ∠𝑍. sin Z= XY XZ = 𝑧 𝑦 Similarly,

cos Z= YZ XZ = 𝑥 𝑦 tan 𝑍= 𝑋𝑌 𝑌𝑍 = 𝑧 𝑥

homework Write the unknown acute angle in the right triangle below in terms of 𝛼 and hence express its sine, cosine and tangent in terms of 𝑚, 𝑛 and 𝑝. 𝛼 𝑝 𝑚 𝑛

Answers to homework The other acute angle is 90°−𝛼 sin 90°−𝛼 = 𝑛 𝑝 cos 90°−𝛼 = 𝑚 𝑝 tan 90°−𝛼 = 𝑛 𝑚

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