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Trigonometric Equations and Identities

Secondary 3 Additional Math practice

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Trigonometric Equations and Identities practice questions

Secondary 3 topics from the Additional Mathematics 4049 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

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What this topic covers

Worked examples

Example 1

Solve 2cos⁡x−1=02\cos x - 1 = 0 for 0∘≤x≤360∘0^\circ \leq x \leq 360^\circ.
Show the worked solution
  • 2cos⁡x=12\cos x = 1
  • cos⁡x=12\cos x = \frac{1}{2}
  • Basic angle=cos⁡−1(12)\text{Basic angle} = \cos^{-1}\left(\frac{1}{2}\right)
  • =60∘= 60^\circ
  • cos is positive in Q1 and Q4\text{cos is positive in Q1 and Q4}
  • x=60∘ or x=360∘−60∘x = 60^\circ \text{ or } x = 360^\circ - 60^\circ
  • x=60∘,300∘x = 60^\circ, 300^\circ

Answer: 60,30060,300

Example 2

Given tan⁡A=815\tan A = \frac{8}{15}, evaluate 1+tan⁡2A1 + \tan^2 A.
Show the worked solution
  • tan⁡A=815\tan A = \frac{8}{15}
  • 1+tan⁡2A=1+642251 + \tan^2 A = 1 + \frac{64}{225}
  • =289225= \frac{289}{225}

Answer: 289/225289/225

Example 3

Given that AA and BB are acute angles, sin⁡A=1237\sin A = \frac{12}{37} and cos⁡B=1213\cos B = \frac{12}{13}, find cos⁡(A−B)\cos(A - B).
Show the worked solution
  • sin⁡A=1237⇒cos⁡A=3537Pythagorean identity\sin A = \frac{12}{37} \Rightarrow \cos A = \frac{35}{37} \quad \scriptsize\textit{Pythagorean identity}
  • cos⁡B=1213⇒sin⁡B=513Pythagorean identity\cos B = \frac{12}{13} \Rightarrow \sin B = \frac{5}{13} \quad \scriptsize\textit{Pythagorean identity}
  • cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A \sin B
  • =3537×1213+1237×513= \frac{35}{37} \times \frac{12}{13} + \frac{12}{37} \times \frac{5}{13}
  • =480481= \frac{480}{481}

Answer: 480/481480/481

Questions parents ask

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Which syllabus does this follow?

Additional Mathematics (4049), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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