find the exact value of cos 75 degrees using sum or difference angle identities
The Correct Answer and Explanation is:
To find the exact value of cos(75°) using sum or difference identities, we express 75° as the sum of two known angles with known cosine and sine values.
We know: 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ75∘=45∘+30∘
Now apply the cosine of sum identity: cos(A+B)=cosAcosB−sinAsinB\cos(A + B) = \cos A \cos B – \sin A \sin Bcos(A+B)=cosAcosB−sinAsinB
Substitute A=45∘A = 45^\circA=45∘ and B=30∘B = 30^\circB=30∘: cos(75∘)=cos(45∘+30∘)=cos45∘cos30∘−sin45∘sin30∘\cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ – \sin 45^\circ \sin 30^\circcos(75∘)=cos(45∘+30∘)=cos45∘cos30∘−sin45∘sin30∘
Recall the exact trigonometric values:
- cos45∘=22\cos 45^\circ = \frac{\sqrt{2}}{2}cos45∘=22
- sin45∘=22\sin 45^\circ = \frac{\sqrt{2}}{2}sin45∘=22
- cos30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}cos30∘=23
- sin30∘=12\sin 30^\circ = \frac{1}{2}sin30∘=21
Substitute these into the identity: cos(75∘)=(22)(32)−(22)(12)\cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) – \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)cos(75∘)=(22)(23)−(22)(21) cos(75∘)=64−24\cos(75^\circ) = \frac{\sqrt{6}}{4} – \frac{\sqrt{2}}{4}cos(75∘)=46−42 cos(75∘)=6−24\cos(75^\circ) = \frac{\sqrt{6} – \sqrt{2}}{4}cos(75∘)=46−2
Final Answer:
cos(75∘)=6−24\boxed{\cos(75^\circ) = \frac{\sqrt{6} – \sqrt{2}}{4}}cos(75∘)=46−2
Explanation:
This problem utilizes a fundamental trigonometric identity to evaluate a non-standard angle. Since 75° is not a common angle on the unit circle, it must be expressed as a sum or difference of angles with known trigonometric values. The cosine of the sum identity provides a reliable method for calculating the exact value. After identifying 75° as 45° + 30°, the identity is applied by substituting in the known values for cosine and sine at those angles. The result simplifies to a rational expression involving square roots, providing the exact trigonometric value of cosine at 75 degrees without needing a calculator.
