Question

defining cofunctions: notation
sine and cosine are cofunctions. that is, they are functions that have equal output values (ratios of side lengths) when their input values (______ measures) are complementary angles.

• $sin(\theta)=cos(\\___ -\theta)$
• $cos(\theta)=sin(90-\theta)$
• $sin(60)=cos(\\___)$

exploring cofunctions
use the property of cofunctions to show the equal trigonometric ratios for triangle abc.
angle a and angle b are complementary angles, because $38^\circ$ plus $52^\circ$ is equal to $90^\circ$.

• $sin(\theta)=cos(90-\theta)$
• $sin(38)=\\_______$
• $cos(\theta)=sin(90-\theta)$
• $cos(38)=\\_______$

Explanation:

Step1: Analyze the first blank (input values)

The input values for sine and cosine (trigonometric functions) are angle measures. So the first blank is "angle".

Step2: Analyze \(\sin(\theta)=\cos(\underline{\quad}-\theta)\)

From the cofunction property, we know that \(\sin(\theta)=\cos(90^{\circ}-\theta)\) (since sine and cosine are cofunctions and complementary angles). So the blank here is \(90^{\circ}\) (or \(90\) if degrees are implied).

Step3: Analyze \(\sin(60^{\circ})=\cos(\underline{\quad})\)

Using \(\sin(\theta)=\cos(90^{\circ}-\theta)\), let \(\theta = 60^{\circ}\). Then \(90^{\circ}-\theta=90^{\circ}- 60^{\circ}=30^{\circ}\). So \(\sin(60^{\circ})=\cos(30^{\circ})\), the blank is \(30^{\circ}\) (or \(30\)).

Step4: Analyze \(\sin(38^{\circ})=\underline{\quad}\)

Using \(\sin(\theta)=\cos(90^{\circ}-\theta)\), let \(\theta = 38^{\circ}\). Then \(90^{\circ}-\theta = 90^{\circ}-38^{\circ}=52^{\circ}\). So \(\sin(38^{\circ})=\cos(52^{\circ})\).

Step5: Analyze \(\cos(38^{\circ})=\underline{\quad}\)

Using \(\cos(\theta)=\sin(90^{\circ}-\theta)\), let \(\theta = 38^{\circ}\). Then \(90^{\circ}-\theta=90^{\circ}-38^{\circ} = 52^{\circ}\). So \(\cos(38^{\circ})=\sin(52^{\circ})\).

Answer:

1. First blank (input values): angle
2. \(\sin(\theta)=\cos(\underline{\quad}-\theta)\): \(90^{\circ}\) (or \(90\))
3. \(\sin(60^{\circ})=\cos(\underline{\quad})\): \(30^{\circ}\) (or \(30\))
4. \(\sin(38^{\circ})=\underline{\quad}\): \(\cos(52^{\circ})\)
5. \(\cos(38^{\circ})=\underline{\quad}\): \(\sin(52^{\circ})\)