Trigonometric Conversion

Geometry Level pending

The expression 4 cos ⁡ x + 3 sin ⁡ x 4\cos { x } +3\sin { x } can be converted to the form C cos ⁡ ( x − θ ) C\cos { (x - \theta ) } . Give the value for C + θ C + \theta .

Give your answer correct to 2 decimal places. Restrict your answer to 0 ≤ θ ≤ π 2 0\le \theta \le \frac { \pi }{ 2 } in radians.


The answer is 5.64.

Geoff Taylor by geoff taylor , 71, Canada

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1 solution

Geoff Taylor
Dec 24, 2017

Process to convert A cos x + B sin x to C cos (x - θ \theta )

C = A 2 + B 2 , A C = cos ⁡ θ , B C = sin ⁡ θ C=\sqrt { { A }^{ 2 }+{ B }^{ 2 } } , \frac { A }{ C } =\cos { \theta } ,\quad \frac { B }{ C } =\sin { \theta }

A c o s x + B s i n x = C ( A C c o s x + B C s i n x ) Acosx\quad +Bsinx\quad =C(\frac { A }{ C } cosx+\frac { B }{ C } sinx)

= C ( c o s θ c o s x + s i n θ s i n x ) = C ( c o s x c o s θ + s i n x s i n θ ) C(cos\theta cos x +sin \theta sin x)=C(cos xcos \theta +sin xsin \theta ) = C c o s ( x − θ ) Ccos(x-\theta )

so in this example A = 4, B = 3 and C = 5

θ = sin ⁡ − 1 3 5 ≈ 0.64 \theta =\sin ^{ -1 }{ \frac { 3 }{ 5 } } \approx 0.64

Thus C + θ ≈ 5.64 C+\theta \quad \approx 5.64

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