Questions: Find the exact value of the expression. Do not use a calculator. sin 29°-cos 61° sin 29°-cos 61°= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the

Solution Steps

To find the exact value of the expression \(\sin 29^{\circ} - \cos 61^{\circ}\), we can use the trigonometric identity that relates sine and cosine functions. Specifically, we know that \(\cos 61^{\circ} = \sin (90^{\circ} - 61^{\circ})\). This simplifies to \(\cos 61^{\circ} = \sin 29^{\circ}\). Therefore, the expression simplifies to \(\sin 29^{\circ} - \sin 29^{\circ}\), which equals 0.

We start by recognizing that \(\cos 61^\circ\) can be rewritten using the co-function identity: \[ \cos 61^\circ = \sin (90^\circ - 61^\circ) = \sin 29^\circ \]

Using the identity from Step 1, we can simplify the given expression: \[ \sin 29^\circ - \cos 61^\circ = \sin 29^\circ - \sin 29^\circ \]

Since \(\sin 29^\circ - \sin 29^\circ = 0\), the expression simplifies to: \[ 0 \]

To ensure the accuracy of our simplification, we calculate the values: \[ \sin 29^\circ \approx 0.4848 \] \[ \cos 61^\circ \approx 0.4848 \] Thus: \[ \sin 29^\circ - \cos 61^\circ \approx 0.4848 - 0.4848 = 0 \]

Final Answer