Question

defining cofunctions: cofunctions
sine and cosine are cofunctions. that is, they are functions that have equal
values (ratios of side lengths) when their input values (angle measures) are
angles.
complementary angles are two angles whose measures sum up to

• $sin(60)=\frac{2sqrt{3}}{4}=\frac{sqrt{quad}}{2}$
• $cos(quad)=\frac{2sqrt{3}}{4}=\frac{sqrt{3}}{2}$
• the two values are equal because they are
• therefore, the sine of $60^circ$ is equal to the cosine of $30^circ$.

Explanation:

Step1: Simplify \(\sin(60^\circ)\)

We have \(\sin(60^\circ)=\frac{2\sqrt{3}}{4}\). Simplify the fraction: \(\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\), so the first blank is \(3\).

Step2: Find the angle for \(\cos\)

We know that \(\cos(30^\circ)=\frac{\sqrt{3}}{2}\) and \(\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\), so the angle in \(\cos(\underline{\quad})\) is \(30^\circ\).

Step3: Identify the relationship

The two values (of \(\sin(60^\circ)\) and \(\cos(30^\circ)\)) are equal because \(60^\circ\) and \(30^\circ\) are complementary angles (their sum is \(90^\circ\)), and sine and cosine are cofunctions, so the last blank is "cofunctions" (or related to complementary angles, but based on the context of cofunctions, the key here is that they are cofunctions or their angles are complementary). Also, complementary angles sum to \(90^\circ\) (or \(\frac{\pi}{2}\) radians, but in degrees it's \(90^\circ\)). For the first part: Sine and cosine are cofunctions, they have equal values when their input angles are complementary. Complementary angles sum to \(90^\circ\).

Filling the blanks:

• First sentence blanks: "values (ratios of side lengths) when their input values (angle measures) are \(\boldsymbol{\text{complementary}}\) angles." (because cofunctions have equal values for complementary angles)
• Complementary angles sum to \(\boldsymbol{90^\circ}\) (or \(\frac{\pi}{2}\) radians, but in degree measure for this triangle, \(90^\circ\))
• \(\sin(60^\circ)=\frac{2\sqrt{3}}{4}=\frac{\sqrt{\boldsymbol{3}}}{2}\)
• \(\cos(\boldsymbol{30^\circ})=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\)
• The two values are equal because they are \(\boldsymbol{\text{cofunctions}}\) (or "of complementary angles", but the key term here is cofunctions or related to complementary angles, and since the last statement says sine of \(60^\circ\) equals cosine of \(30^\circ\) because they are cofunctions for complementary angles)

Answer:

• First blank (in \(\sin(60)=\frac{2\sqrt{3}}{4}=\frac{\sqrt{\underline{}}}{2}\)): \(3\)
• Angle in \(\cos(\underline{\quad})\): \(30^\circ\)
• Complementary angles sum to: \(90^\circ\) (or \(\frac{\pi}{2}\) radians, but \(90^\circ\) here)
• Relationship: cofunctions (or "of complementary angles")
• First part blanks: "complementary" (for the angle relationship of cofunctions)

(For the specific blanks in the text:

1. Sine and cosine are cofunctions. That is, they are functions that have equal \(\boldsymbol{\text{trigonometric}}\) (or "function") values (ratios of side lengths) when their input values (angle measures) are \(\boldsymbol{\text{complementary}}\) angles.
2. Complementary angles are two angles whose measures sum up to \(\boldsymbol{90^\circ}\) (or \(\frac{\pi}{2}\) radians).
3. \(\sin(60)=\frac{2\sqrt{3}}{4}=\frac{\sqrt{\boldsymbol{3}}}{2}\)
4. \(\cos(\boldsymbol{30^\circ})=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\)
5. The two values are equal because they are \(\boldsymbol{\text{cofunctions}}\) (or "of complementary angles", but the context is cofunctions, so "cofunctions" or "for complementary angles") )