Use the conversion formulas to convert from polar coordinates to rectangular coordinates.

Substitute in the known values of <math><mstyle displaystyle="true"><mi>r</mi><mo>=</mo><mn>1.5</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> into the formulas.

Add full rotations of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> until the angle is greater than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and less than <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

The exact value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Move the leading negative in <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> into the numerator.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>1.5</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.75</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>0.75</mn></mstyle></math> by <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Add full rotations of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> until the angle is greater than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and less than <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>1.5</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The rectangular representation of the polar point <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1.5</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>1.2990381</mn><mo>,</mo><mn>0.75</mn><mo>)</mo></mrow></mstyle></math> .

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