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Note: Figures not drawn to scale. $\newline$ The angles shown above are acute and $\newline$ $\sin(a^{\circ})=\cos(b^{\circ})$ . If $\newline$ $a=4k-22$ and $\newline$ $b=6k-13$ , what is the value of $\newline$ $k$ ?

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Evaluate expression when a complex numbers and a variable term is given

Full solution

Q. Note: Figures not drawn to scale.

\newline

The angles shown above are acute and

\newline

\sin(a^{\circ})=\cos(b^{\circ})

. If

\newline

a=4k-22

and

\newline

b=6k-13

, what is the value of

\newline

k

?

Identify Complementary Angles: Given that $\sin(a^{\circ})=\cos(b^{\circ})$ and the angles are acute, we can use the complementary angle identity which states that $\sin(\theta) = \cos(90 - \theta)$ for acute angles. This means that $a$ and $b$ are complementary angles.
Write Equation for Sum: Since $a$ and $b$ are complementary, we can write the equation $a + b = 90$ degrees. $\newline$ Substitute the given expressions for $a$ and $b$ into this equation: $(4k - 22) + (6k - 13) = 90$ .
Combine Like Terms: Combine like terms: $4k + 6k - 22 - 13 = 90$ . This simplifies to $10k - 35 = 90$ .
Isolate Term with k: Add $35$ to both sides of the equation to isolate the term with $k$ : $10k - 35 + 35 = 90 + 35$ . This simplifies to $10k = 125$ .
Solve for $k$ : Divide both sides of the equation by $10$ to solve for $k$ : $\frac{10k}{10} = \frac{125}{10}$ . This simplifies to $k = 12.5$ .

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