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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$\begingroup$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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