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Chapter 6: Q. 87 (page 381)

Find the exact value of: sin40°+sin130°+sin220°+sin310°

Short Answer

Expert verified

The value ofsin40°+sin130°+sin220°+sin310°=0

Step by step solution

01

Step 1. Given information

The given value issin40°+sin130°+sin220°+sin310°

02

Step 2. Sketch the graph of sin130°for reference angle α

The reference angle for sin130°is50°

03

Step 3. Sketch the graph of sin220° for reference angle

The reference angle for sin220°is40°

04

Step 4. Sketch the graph of sin310°for reference angle

The reference angle for sin310°is50°

05

Step 5. Finding the value of sin40°+sin130°+sin220°+sin310°

Ifαis any reference angle forθthen,

sinθ=±sinα

so,sin130°=±sin50°,sin220°=±sin40°,andsin310°=±sin50°

Now check in which quadrant the given angles lie

sin40° lies in quadrant-I,where is positive

sin130° lies in quadrant-II,where is positive

sin220°lies in quadrant-III,where is negative

sin310° lies in quadrant-IV,where is negative

Replace the given angle in the expression with the respective reference angle

sin40°+sin130°+sin220°+sin310°=sin40°+sin50°+(-sin40°)+(-sin50°)=(sin40°-sin40°)+(sin50°-sin50°)=0

Therefore,the value ofsin40°+sin130°+sin220°+sin310°=0

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