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Can someone please walk me through the steps on how I can get an answer for a decimal problem involving sum and difference formulas?

Thank you very much, Jeremy

If $\text{cos}(\alpha) = 0.167$ and $\text{sin}(\beta) = 0.529$ with both angles’ terminal rays in Quadrant-$I$, find the values of $$\sin(\alpha+\beta) \ \ \text{and}\ \cos(\alpha-\beta)$$ Your answers should be accurate to $4$ decimal places.

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  • $\begingroup$ In the context of the problem, do you have trig tables? $\endgroup$ – quasi Nov 19 '17 at 2:04
  • $\begingroup$ Use your trigonometric identities, they may help. $\endgroup$ – Landuros Nov 19 '17 at 2:05
  • $\begingroup$ The OP said: "without a calculator". My guess is trig tables are allowed, so just find the angles by reverse lookup, add them, and look up the trig values for the sum of the angles. $\endgroup$ – quasi Nov 19 '17 at 2:05
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    $\begingroup$ @quasi That method still involves square roots, doesn't it? It will be a tough question to solve without a calculator. $\endgroup$ – Landuros Nov 19 '17 at 2:13
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    $\begingroup$ The very nature of the statement of the problem strongly suggests that either calculators are allowed, or else trig tables are available. $\endgroup$ – quasi Nov 19 '17 at 2:16
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Well then, if you have a calculator, then this becomes relatively simple. I'm sure quasi has already answered the question in the comments, but I'd just like to properly answer it. Three trigonometric identities will be particularly useful in this question.

$\sin^2(\theta)+\cos^2(\theta)\equiv1$

$\sin(α + β) \equiv \sin(α) \cos(β) + \cos(α) \sin(β)$

$\cos(α – β) \equiv \cos(α) \cos(β) + \sin(α) \sin(β)$

Given $\sin(β) = 0.529$ and $\cos(α)=0.167$. I assume we are working in radians.

Find $\sin(α)$:

\begin{align} \sin^2(\theta) + \cos^2(\theta) & \equiv 1 \\ \sin^2(α) + \cos^2(α) &= 1 \\ \sin^2(α) + 0.027889 &= 1 \\ \sin^2(α) &= 1 - 0.027889 \\ \sin^2(α) &= 0.972111 \\ \sin(α) & = 0.98596 \space(\text{to 5 d.p., reject} \sin(α) \lt 0) \end{align}

Similarly, we find that $\cos(β) = 0.84862.$

Then we plug those values into the identity.

\begin{align} \sin(α + β) & \equiv \sin(α) \cos(β) + \cos(α) \sin(β) \\ \sin(α + β) & = 0.98596 \cdot 0.84862 + 0.167 \cdot 0.529 \\ & = 0.9250 \space(\text{to 4 d.p.}) \end{align}

\begin{align} \cos(α \space – β) & \equiv \cos(α) \cos(β) + \sin(α) \sin(β) \\ \cos(α \space – β) & = 0.167 \cdot 0.84862 + 0.98596 \cdot 0.529 \\ & = 0.6633 \space(\text{to 4 d.p.}) \end{align}

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  • $\begingroup$ As the input was only three decimal places it needs justification that your values are accurate to four. Since the actual values are close to $1$, that may be true but I am doubtful about the cosine being that good. That said, this is clearly the intended approach. $\endgroup$ – Ross Millikan Nov 19 '17 at 3:20
  • $\begingroup$ @RossMillikan I agree, this isn't the best question for an exam/test. But with exact values, it would be easier, as the OP has stated. $\endgroup$ – Landuros Nov 19 '17 at 7:20

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