The form R sin(θ + α) Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level The form R sin(θ + α) questions. See exactly how to solve problems on R sin form, harmonic form.

R sin formharmonic form
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Express 3sin⁡θ+4cos⁡θ3\sin\theta + 4\cos\theta in the form Rsin⁡(θ+α)R\sin(\theta+\alpha) where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ. State the exact value of RR.

Worked solution

  1. Set up the identity for the harmonic form

    3sin⁡θ+4cos⁡θ=Rsin⁡(θ+α)3\sin\theta + 4\cos\theta=R\sin(\theta+\alpha)

    Introduce R and alpha to be found.

  2. Compare coefficients of sin and cos

    Rcos⁡α=3, Rsin⁡α=4R\cos\alpha=3,\ R\sin\alpha=4

    Matching each term gives two equations for R and alpha.

  3. Find R by squaring and adding the coefficients

    R=25=5R=\sqrt{25}=5

    R = sqrt(a^2+b^2) as an exact surd.

Answer
55
Question 2
2 markseasy
Express 5sin⁡θ+12cos⁡θ5\sin\theta + 12\cos\theta in the form Rsin⁡(θ+α)R\sin(\theta+\alpha) where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ. State the exact value of RR.

Worked solution

  1. Set up the identity for the harmonic form

    5sin⁡θ+12cos⁡θ=Rsin⁡(θ+α)5\sin\theta + 12\cos\theta=R\sin(\theta+\alpha)

    Introduce R and alpha to be found.

  2. Compare coefficients of sin and cos

    Rcos⁡α=5, Rsin⁡α=12R\cos\alpha=5,\ R\sin\alpha=12

    Matching each term gives two equations for R and alpha.

  3. Find R by squaring and adding the coefficients

    R=169=13R=\sqrt{169}=13

    R = sqrt(a^2+b^2) as an exact surd.

Answer
1313
Question 3
2 markseasy
Express 8sin⁡θ+15cos⁡θ8\sin\theta + 15\cos\theta in the form Rsin⁡(θ+α)R\sin(\theta+\alpha) where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ. State the exact value of RR.

Worked solution

  1. Set up the identity for the harmonic form

    8sin⁡θ+15cos⁡θ=Rsin⁡(θ+α)8\sin\theta + 15\cos\theta=R\sin(\theta+\alpha)

    Introduce R and alpha to be found.

  2. Compare coefficients of sin and cos

    Rcos⁡α=8, Rsin⁡α=15R\cos\alpha=8,\ R\sin\alpha=15

    Matching each term gives two equations for R and alpha.

  3. Find R by squaring and adding the coefficients

    R=289=17R=\sqrt{289}=17

    R = sqrt(a^2+b^2) as an exact surd.

Answer
1717
Question 4
2 markseasy
Express 3sin⁡θ+4cos⁡θ3\sin\theta + 4\cos\theta in the form Rsin⁡(θ+α)R\sin(\theta+\alpha) where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ. Find α\alpha in degrees to 1 d.p.

Worked solution

  1. Set up the identity for the harmonic form

    3sin⁡θ+4cos⁡θ=Rsin⁡(θ+α)3\sin\theta + 4\cos\theta=R\sin(\theta+\alpha)

    Introduce R and alpha to be found.

  2. Compare coefficients of sin and cos

    Rcos⁡α=3, Rsin⁡α=4R\cos\alpha=3,\ R\sin\alpha=4

    Matching each term gives two equations for R and alpha.

  3. Find alpha by dividing the coefficients

    tan⁡α=43 ⇒ α=53.1∘\tan\alpha=\frac{4}{3}\ \Rightarrow\ \alpha=53.1^\circ

    alpha = arctan(b/a).

Answer
53.1∘53.1^\circ
Question 5
2 markseasy
Express 4sin⁡θ+3cos⁡θ4\sin\theta + 3\cos\theta in the form Rsin⁡(θ+α)R\sin(\theta+\alpha) where R>0R>0 and 0∘<α<90∘0^\circ<\alpha<90^\circ. Find α\alpha in degrees to 1 d.p.

Worked solution

  1. Set up the identity for the harmonic form

    4sin⁡θ+3cos⁡θ=Rsin⁡(θ+α)4\sin\theta + 3\cos\theta=R\sin(\theta+\alpha)

    Introduce R and alpha to be found.

  2. Compare coefficients of sin and cos

    Rcos⁡α=4, Rsin⁡α=3R\cos\alpha=4,\ R\sin\alpha=3

    Matching each term gives two equations for R and alpha.

  3. Find alpha by dividing the coefficients

    tan⁡α=34 ⇒ α=36.9∘\tan\alpha=\frac{3}{4}\ \Rightarrow\ \alpha=36.9^\circ

    alpha = arctan(b/a).

Answer
36.9∘36.9^\circ

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