Question

Question was provided via image upload.

Explanation:

Step1: Fill the cosine angle

In right triangle \( XYZ \) with \( \angle Z = 90^\circ \) and \( \angle X = 23^\circ \), for the cosine ratio, \( \cos(\angle X)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{XZ}{XY} \). So the angle in \( \cos(\underline{\quad}^\circ)=\frac{XZ}{26} \) is \( 23^\circ \).

Step2: Solve for \( YZ \) in \( (26)(\sin(23^\circ)) = \underline{\quad} \)

We know \( \sin(23^\circ)=\frac{YZ}{26} \), so cross - multiplying gives \( YZ = 26\times\sin(23^\circ) \), so the blank here is \( YZ \).

Step3: Fill the coefficient for \( 0.3907 \)

Since \( YZ = 26\times\sin(23^\circ) \) and \( \sin(23^\circ)\approx0.3907 \), the coefficient is \( 26 \), so \( (26)(0.3907)=YZ \).

Step4: Calculate \( (26)(0.9205) \)

We know that \( \cos(23^\circ)\approx0.9205 \) and \( XZ = 26\times\cos(23^\circ) \), so \( 26\times0.9205 = 23.933 \approx23.9 \) (rounded to the nearest tenth).

Answer:

• For \( \cos(\underline{\quad}^\circ)=\frac{XZ}{26} \), the answer is \( 23 \).
• For \( (26)(\sin(23^\circ)) = \underline{\quad} \), the answer is \( YZ \).
• For \( (\underline{\quad})(0.3907)=YZ \), the answer is \( 26 \).
• For \( (26)(0.9205)=\underline{\quad} \), the answer is \( 23.9 \).
• \( m\angle Y = 67^\circ \), \( YZ\approx10.2 \) (rounded to the nearest tenth, \( 10.159\approx10.2 \)), \( XZ\approx23.9 \)