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Solve for x in each of the following questions. Give your answers correct to one decimal place, where 0^ exto ext ext extless= ext x ext ext extless= ext 90^ exto. Prove that riangle OAC ext ext extde

Mathematics

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Answers

3 parts
1.1

34.1∘34.1^\circ

1.2

63.4∘63.4^\circ

4
Shown:

△OAC≡△OBC\triangle OAC \equiv \triangle OBC

Step-by-step solution

9 steps

here are the solutions for Questions 1 and 4.

Part 1

Solve for xx in each of the following questions. Give your answers correct to one decimal place, where 0∘≤x≤90∘0^\circ \le x \le 90^\circ.

Part 1.1

34.1∘34.1^\circ

2sin⁡x=1.122\sin x = 1.12

  1. Isolate sin⁡x\sin x.

    sin⁡x=1.122\sin x = \frac{1.12}{2} sin⁡x=0.56\sin x = 0.56

  2. Use the inverse sine function to find xx.

    x=arcsin⁡(0.56)x = \arcsin(0.56) x≈34.06∘x \approx 34.06^\circ

  3. Round to one decimal place.

    x=34.1∘x = \boxed{34.1^\circ}

Part 1.2

63.4∘63.4^\circ

tan⁡x=4cos⁡60∘\tan x = 4\cos 60^\circ

  1. Evaluate cos⁡60∘\cos 60^\circ.

    We know that cos⁡60∘=12\cos 60^\circ = \frac{1}{2}.

  2. Substitute the value and simplify the right side.

    tan⁡x=4×12\tan x = 4 \times \frac{1}{2} tan⁡x=2\tan x = 2

  3. Use the inverse tangent function to find xx.

    x=arctan⁡(2)x = \arctan(2) x≈63.43∘x \approx 63.43^\circ

  4. Round to one decimal place.

    x=63.4∘x = \boxed{63.4^\circ}

Part 4

△OAC≡△OBC\triangle OAC \equiv \triangle OBC

Prove that △OAC≡△OBC\triangle OAC \equiv \triangle OBC

  1. Identify common and equal sides.

    • OA=OBOA = OB (Radii of the same circle)
    • OC=OCOC = OC (Common side)
    • AC=BCAC = BC (Given by the tick marks in the diagram)

  2. Apply the Side-Side-Side (SSS) congruence criterion.

    Since all three corresponding sides of △OAC\triangle OAC and △OBC\triangle OBC are equal, the triangles are congruent by the SSS criterion.
    Therefore, △OAC≡△OBC (SSS)\boxed{\triangle OAC \equiv \triangle OBC \text{ (SSS)}}.

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