Answer :
Sure, let's match each sine or cosine value to its equivalent measure step by step.
### Step 1: Find the Equivalent Angle for \( \cos(447^\circ) \)
To find the equivalent measure of \( \cos(447^\circ) \) within the standard range [0, 360), we perform a modulo operation:
[tex]\[ 447^\circ \mod 360^\circ = 87^\circ \][/tex]
So, \( \cos(447^\circ) = \cos(87^\circ) \).
### Step 2: Calculate and Recognize Standard Trigonometric Values
Here are the trigonometric values for the given angles:
- \( \cos(87^\circ) \approx 0.052336 \)
- \( \cos(58^\circ) \approx 0.529919 \)
- \( \sin(33^\circ) \approx 0.544639 \)
- \( \cos(137^\circ) \approx -0.731354 \)
- \( \sin(123^\circ) \approx 0.838671 \)
- \( \cos(74^\circ) \approx 0.275637 \)
### Step 3: Match Each Value
Let's match each given trigonometric function with its calculated value:
1. \( \cos(447^\circ) = \cos(87^\circ) \approx 0.052336 \)
2. \( \cos(58^\circ) \approx 0.529919 \)
3. \( \sin(33^\circ) \approx 0.544639 \)
4. \( \cos(137^\circ) \approx -0.731354 \)
5. \( \sin(123^\circ) \approx 0.838671 \)
6. \( \cos(74^\circ) \approx 0.275637 \)
### Step 4: Place the Values in the Blanks
Now we can fill in the blanks with the corresponding values:
[tex]\[ \begin{array}{cccccccc} \cos(447^\circ) & \cos(58^\circ) & \sin(33^\circ) & \cos(137^\circ) & \sin(123^\circ) & & & \cos(74^\circ) \\ 0.052336 & 0.529919 & 0.544639 & -0.731354 & 0.838671 & & & 0.275637 \\ \end{array} \][/tex]
### Final Matching Representation
[tex]\[ \cos(447^\circ) \longleftrightarrow 0.052336 \][/tex]
[tex]\[ \cos(58^\circ) \longleftrightarrow 0.529919 \][/tex]
[tex]\[ \sin(33^\circ) \longleftrightarrow 0.544639 \square \cos(137^\circ) \][/tex]
[tex]\[ \sin(123^\circ) \longleftrightarrow 0.838671 \square \cos(74^\circ) \][/tex]
In words:
- \( \cos(447^\circ) \leftrightarrow 0.052336 \)
- \( \cos(58^\circ) \leftrightarrow 0.529919 \)
- \( \sin(33^\circ) \leftrightarrow 0.544639 \)
- \( \cos(137^\circ) \leftrightarrow -0.731354 \)
- \( \sin(123^\circ) \leftrightarrow 0.838671 \)
- [tex]\( \cos(74^\circ) \leftrightarrow 0.275637 \)[/tex]
### Step 1: Find the Equivalent Angle for \( \cos(447^\circ) \)
To find the equivalent measure of \( \cos(447^\circ) \) within the standard range [0, 360), we perform a modulo operation:
[tex]\[ 447^\circ \mod 360^\circ = 87^\circ \][/tex]
So, \( \cos(447^\circ) = \cos(87^\circ) \).
### Step 2: Calculate and Recognize Standard Trigonometric Values
Here are the trigonometric values for the given angles:
- \( \cos(87^\circ) \approx 0.052336 \)
- \( \cos(58^\circ) \approx 0.529919 \)
- \( \sin(33^\circ) \approx 0.544639 \)
- \( \cos(137^\circ) \approx -0.731354 \)
- \( \sin(123^\circ) \approx 0.838671 \)
- \( \cos(74^\circ) \approx 0.275637 \)
### Step 3: Match Each Value
Let's match each given trigonometric function with its calculated value:
1. \( \cos(447^\circ) = \cos(87^\circ) \approx 0.052336 \)
2. \( \cos(58^\circ) \approx 0.529919 \)
3. \( \sin(33^\circ) \approx 0.544639 \)
4. \( \cos(137^\circ) \approx -0.731354 \)
5. \( \sin(123^\circ) \approx 0.838671 \)
6. \( \cos(74^\circ) \approx 0.275637 \)
### Step 4: Place the Values in the Blanks
Now we can fill in the blanks with the corresponding values:
[tex]\[ \begin{array}{cccccccc} \cos(447^\circ) & \cos(58^\circ) & \sin(33^\circ) & \cos(137^\circ) & \sin(123^\circ) & & & \cos(74^\circ) \\ 0.052336 & 0.529919 & 0.544639 & -0.731354 & 0.838671 & & & 0.275637 \\ \end{array} \][/tex]
### Final Matching Representation
[tex]\[ \cos(447^\circ) \longleftrightarrow 0.052336 \][/tex]
[tex]\[ \cos(58^\circ) \longleftrightarrow 0.529919 \][/tex]
[tex]\[ \sin(33^\circ) \longleftrightarrow 0.544639 \square \cos(137^\circ) \][/tex]
[tex]\[ \sin(123^\circ) \longleftrightarrow 0.838671 \square \cos(74^\circ) \][/tex]
In words:
- \( \cos(447^\circ) \leftrightarrow 0.052336 \)
- \( \cos(58^\circ) \leftrightarrow 0.529919 \)
- \( \sin(33^\circ) \leftrightarrow 0.544639 \)
- \( \cos(137^\circ) \leftrightarrow -0.731354 \)
- \( \sin(123^\circ) \leftrightarrow 0.838671 \)
- [tex]\( \cos(74^\circ) \leftrightarrow 0.275637 \)[/tex]
