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2 years ago

Find the value of x?

Mathematics

1 answer:

Deffense [45]2 years ago

6 0

Answer:

145°

Step-by-step explanation:

See the attachment. Angles A, B, and C all belong to the triangle.

We know that A + B + C = 180° since the angles of a triangle always add to 180°. We also know that when two lines meet that their sum is also equal to 180°. So we can write for each interior angle the following:

Result

<A = (180 - 88) 72

<B = (180-x) 180-x

<C = (180 - 127) 53

The sum of angles A, B, and C is equal to 180:

72 + (180-x) + 53 = 180

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Answer: see proof below

Step-by-step explanation:

Use the Double Angle Identity: sin 2Ф = 2sinФ · cosФ

Use the Sum/Difference Identities:

sin(α + β) = sinα · cosβ + cosα · sinβ

cos(α - β) = cosα · cosβ + sinα · sinβ

Use the Unit circle to evaluate: sin45 = cos45 = √2/2

Use the Double Angle Identities: sin2Ф = 2sinФ · cosФ

Use the Pythagorean Identity: cos²Ф + sin²Ф = 1

Proof LHS → RHS

LHS: 2sin(45 + 2A) · cos(45 - 2A)

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Unit Circle: 2[(√2/2)cos2A + (√2/2)sin2A][(√2/2)cos2A +(√2/2)·sin2A)]

Expand: 2[(1/2)cos²2A + cos2A·sin2A + (1/2)sin²2A]

Distribute: cos²2A + 2cos2A·sin2A + sin²2A

Pythagorean Identity: 1 + 2cos2A·sin2A

Double Angle: 1 + sin4A

LHS = RHS: 1 + sin4A = 1 + sin4A $\checkmark$

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