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Sum and Difference Identities

Show your work for each problem.

  1. 1.
    Which expression is equal to sin⁡(A+B)\sin(A + B)?
    1. a)sin⁡Asin⁡B+cos⁡Acos⁡B\sin A \sin B + \cos A \cos B
    2. b)cos⁡Acos⁡B−sin⁡Asin⁡B\cos A \cos B - \sin A \sin B
    3. c)sin⁡Acos⁡B+cos⁡Asin⁡B\sin A \cos B + \cos A \sin B
    4. d)sin⁡A+sin⁡B\sin A + \sin B

    Answer: sin⁡Acos⁡B+cos⁡Asin⁡B\sin A \cos B + \cos A \sin B

    sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B. The option sin⁡A+sin⁡B\sin A + \sin B is the classic mistake, and cos⁡Acos⁡B−sin⁡Asin⁡B\cos A \cos B - \sin A \sin B is cos⁡(A+B)\cos(A + B).

  2. 2.
    Simplify without a calculator: sin⁡70∘cos⁡20∘+cos⁡70∘sin⁡20∘\sin 70^\circ \cos 20^\circ + \cos 70^\circ \sin 20^\circ

    Answer: 1

    This is the sine sum identity backwards: sin⁡(70∘+20∘)=sin⁡90∘=1\sin(70^\circ + 20^\circ) = \sin 90^\circ = 1.

  3. 3.
    Use the cosine difference identity to find cos⁡(90∘−30∘)\cos(90^\circ - 30^\circ).

    Answer: 12\frac{1}{2}

    1. What is cos⁡90∘\cos 90^\circ? 0
    2. What is sin⁡90∘\sin 90^\circ? 1
    3. Compute cos⁡90∘cos⁡30∘+sin⁡90∘sin⁡30∘\cos 90^\circ \cos 30^\circ + \sin 90^\circ \sin 30^\circ. 1/2
  4. 4.
    Which expression is equal to cos⁡(A−B)\cos(A - B)?
    1. a)sin⁡Acos⁡B−cos⁡Asin⁡B\sin A \cos B - \cos A \sin B
    2. b)cos⁡Acos⁡B−sin⁡Asin⁡B\cos A \cos B - \sin A \sin B
    3. c)cos⁡Acos⁡B+sin⁡Asin⁡B\cos A \cos B + \sin A \sin B
    4. d)cos⁡A−cos⁡B\cos A - \cos B

    Answer: cos⁡Acos⁡B+sin⁡Asin⁡B\cos A \cos B + \sin A \sin B

    cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A \sin B. With a minus sign it would be cos⁡(A+B)\cos(A + B), and sin⁡Acos⁡B−cos⁡Asin⁡B\sin A \cos B - \cos A \sin B is sin⁡(A−B)\sin(A - B).

  5. 5.
    Simplify without a calculator: cos⁡50∘cos⁡10∘−sin⁡50∘sin⁡10∘\cos 50^\circ \cos 10^\circ - \sin 50^\circ \sin 10^\circ

    Answer: 1/2

    cos⁡Acos⁡B−sin⁡Asin⁡B=cos⁡(A+B)\cos A \cos B - \sin A \sin B = \cos(A + B), so the expression is cos⁡(50∘+10∘)=cos⁡60∘=12\cos(50^\circ + 10^\circ) = \cos 60^\circ = \frac{1}{2}.

  6. 6.
    Find sin⁡105∘\sin 105^\circ with 105∘=60∘+45∘105^\circ = 60^\circ + 45^\circ.

    Answer: 6+24\frac{\sqrt{6} + \sqrt{2}}{4}

    1. Compute sin⁡60∘cos⁡45∘=32⋅22\sin 60^\circ \cos 45^\circ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} (exact, like sqrt(6)/4, or 4 decimal places). 0.6124
    2. Compute cos⁡60∘sin⁡45∘=12⋅22\cos 60^\circ \sin 45^\circ = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} (exact or 4 decimal places). 0.3536
    3. Add the exact products 64+24\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} and round only at the end (adding the rounded products gives 0.9660). Give sin⁡105∘\sin 105^\circ rounded to 4 decimal places. 0.9659
  7. 7.
    What is the exact value of sin⁡75∘\sin 75^\circ?
    1. a)3+14\frac{\sqrt{3} + 1}{4}
    2. b)12+22\frac{1}{2} + \frac{\sqrt{2}}{2}
    3. c)6−24\frac{\sqrt{6} - \sqrt{2}}{4}
    4. d)6+24\frac{\sqrt{6} + \sqrt{2}}{4}

    Answer: 6+24\frac{\sqrt{6} + \sqrt{2}}{4}

    sin⁡75∘=sin⁡45∘cos⁡30∘+cos⁡45∘sin⁡30∘=64+24=6+24≈0.9659\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659. The option 12+22\frac{1}{2} + \frac{\sqrt{2}}{2} is sin⁡30∘+sin⁡45∘≈1.21\sin 30^\circ + \sin 45^\circ \approx 1.21, more than 1, so it cannot be a sine.

  8. 8.
    Use tan⁡(45∘−30∘)\tan(45^\circ - 30^\circ) with tan⁡45∘=1\tan 45^\circ = 1 and tan⁡30∘=33\tan 30^\circ = \frac{\sqrt{3}}{3} to find tan⁡15∘\tan 15^\circ. Round to 4 decimal places.

    Answer: 0.2679

    tan⁡15∘=1−331+33=3−33+3=2−3≈0.2679\tan 15^\circ = \frac{1 - \frac{\sqrt{3}}{3}}{1 + \frac{\sqrt{3}}{3}} = \frac{3 - \sqrt{3}}{3 + \sqrt{3}} = 2 - \sqrt{3} \approx 0.2679.

  9. 9.
    sin⁡A=513\sin A = \frac{5}{13} with AA in Quadrant I, and cos⁡B=−35\cos B = -\frac{3}{5} with BB in Quadrant II. Find sin⁡(A+B)\sin(A + B).

    Answer: 3365\frac{33}{65}

    1. What is cos⁡A\cos A? 12/13
    2. What is sin⁡B\sin B? 4/5
    3. Compute sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B as a fraction. 33/65
  10. 10.
    Find cos⁡75∘\cos 75^\circ with 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ.

    Answer: 6−24\frac{\sqrt{6} - \sqrt{2}}{4}

    1. Compute cos⁡45∘cos⁡30∘=22⋅32\cos 45^\circ \cos 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} (exact or 4 decimal places). 0.6124
    2. Compute sin⁡45∘sin⁡30∘=22⋅12\sin 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} (exact or 4 decimal places). 0.3536
    3. Subtract the second product from the first. Give cos⁡75∘\cos 75^\circ rounded to 4 decimal places. 0.2588
  11. 11.
    Which value is equal to cos⁡80∘cos⁡20∘+sin⁡80∘sin⁡20∘\cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ?
    1. a)cos⁡60∘\cos 60^\circ
    2. b)sin⁡100∘\sin 100^\circ
    3. c)cos⁡100∘\cos 100^\circ
    4. d)sin⁡60∘\sin 60^\circ

    Answer: cos⁡60∘\cos 60^\circ

    cos⁡Acos⁡B+sin⁡Asin⁡B=cos⁡(A−B)\cos A \cos B + \sin A \sin B = \cos(A - B), so the expression is cos⁡(80∘−20∘)=cos⁡60∘\cos(80^\circ - 20^\circ) = \cos 60^\circ. Choosing cos⁡100∘\cos 100^\circ keeps the sign by mistake.

  12. 12.
    sin⁡A=45\sin A = \frac{4}{5} and cos⁡B=1213\cos B = \frac{12}{13}, with AA and BB in Quadrant I. Find cos⁡(A+B)\cos(A + B) as a fraction.

    Answer: 16/65

    cos⁡(A+B)=35⋅1213−45⋅513=3665−2065=1665\cos(A + B) = \frac{3}{5} \cdot \frac{12}{13} - \frac{4}{5} \cdot \frac{5}{13} = \frac{36}{65} - \frac{20}{65} = \frac{16}{65}.

  13. 13.
    tan⁡A=2\tan A = 2 and tan⁡B=3\tan B = 3, with AA and BB acute angles. Find tan⁡(A+B)\tan(A + B) and the angle A+BA + B.

    Answer: tan⁡(A+B)=−1\tan(A + B) = -1, A+B=135∘A + B = 135^\circ

    1. What is the numerator tan⁡A+tan⁡B\tan A + \tan B? 5
    2. What is the denominator 1−tan⁡Atan⁡B1 - \tan A \tan B? -5
    3. What is tan⁡(A+B)\tan(A + B)? -1
    4. A+BA + B lies between 0∘0^\circ and 180∘180^\circ. Which angle there has tangent −1-1? (in degrees) 135
  14. 14.
    sin⁡A=35\sin A = \frac{3}{5} with AA in Quadrant I. Use cos⁡(A+B)\cos(A + B) with B=AB = A to find cos⁡2A\cos 2A.

    Answer: 725\frac{7}{25}

    1. What is cos⁡A\cos A? 4/5
    2. What is cos⁡Acos⁡A\cos A \cos A? 16/25
    3. What is sin⁡Asin⁡A\sin A \sin A? 9/25
    4. What is cos⁡2A=cos⁡Acos⁡A−sin⁡Asin⁡A\cos 2A = \cos A \cos A - \sin A \sin A? 7/25
  15. 15.
    What is the exact value of cos⁡165∘\cos 165^\circ? (Hint: 165∘=120∘+45∘165^\circ = 120^\circ + 45^\circ)
    1. a)6−24\frac{\sqrt{6} - \sqrt{2}}{4}
    2. b)6+24\frac{\sqrt{6} + \sqrt{2}}{4}
    3. c)2−64\frac{\sqrt{2} - \sqrt{6}}{4}
    4. d)−6+24-\frac{\sqrt{6} + \sqrt{2}}{4}

    Answer: −6+24-\frac{\sqrt{6} + \sqrt{2}}{4}

    cos⁡165∘=cos⁡120∘cos⁡45∘−sin⁡120∘sin⁡45∘=−24−64=−6+24≈−0.9659\cos 165^\circ = \cos 120^\circ \cos 45^\circ - \sin 120^\circ \sin 45^\circ = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4} \approx -0.9659. It must be negative and close to −1-1, because 165∘165^\circ is close to 180∘180^\circ.

  16. 16.
    cos⁡A=−35\cos A = -\frac{3}{5} with AA in Quadrant II, and sin⁡B=−513\sin B = -\frac{5}{13} with BB in Quadrant III. Find sin⁡(A−B)\sin(A - B) as a fraction.

    Answer: -63/65

    sin⁡A=45\sin A = \frac{4}{5} and cos⁡B=−1213\cos B = -\frac{12}{13}. sin⁡(A−B)=45⋅(−1213)−(−35)⋅(−513)=−4865−1565=−6365\sin(A - B) = \frac{4}{5} \cdot \left(-\frac{12}{13}\right) - \left(-\frac{3}{5}\right) \cdot \left(-\frac{5}{13}\right) = -\frac{48}{65} - \frac{15}{65} = -\frac{63}{65}.

  17. 17.
    cos⁡A=−45\cos A = -\frac{4}{5} with AA in Quadrant III, and sin⁡B=1213\sin B = \frac{12}{13} with BB in Quadrant II. Find cos⁡(A+B)\cos(A + B).

    Answer: 5665\frac{56}{65}

    1. What is sin⁡A\sin A? -3/5
    2. What is cos⁡B\cos B? -5/13
    3. Compute cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B as a fraction. 56/65
  18. 18.
    Find tan⁡105∘\tan 105^\circ with 105∘=60∘+45∘105^\circ = 60^\circ + 45^\circ.

    Answer: −2−3-2 - \sqrt{3}

    1. What is the numerator tan⁡60∘+tan⁡45∘\tan 60^\circ + \tan 45^\circ? (exact, like sqrt(3)+1, or 4 decimal places) 2.7321
    2. What is the denominator 1−tan⁡60∘tan⁡45∘1 - \tan 60^\circ \tan 45^\circ (exact or 4 decimal places)? -0.7321
    3. Divide the exact values 3+11−3\frac{\sqrt{3} + 1}{1 - \sqrt{3}} and round only at the end (dividing the rounded values gives -3.7319). Give tan⁡105∘\tan 105^\circ rounded to 4 decimal places. -3.7321
Interactive lesson with step-by-step solutions and more practice (free): mathorio.com/en/learn/trigonometry/identities/sum-difference-identities
Today's challengeOne problem, three tries