Splitting angles to conquer them

Mathematics · Geometry · ages 16-17

Name ______________________ Date ____________

To use special angles for cos 15°, write 15° as 45° - k°. What is k?
Answer: ______________
Which split finds cos 15° from special angles?
- 45° - 30°
- 60° - 30°
- 45° - 15°
- 90° - 60°
Which expansion of sin of A plus B is correct?
- Sin A sin B plus cos A cos B
- Sin A cos B plus cos A sin B
- Sin A cos B minus cos A sin B
Cosine expansions keep the middle sign, just like sine.
Circle one: True False
What is sin(90° − 60°)?
Answer: ______________
After a 30 degree shift, the interval 0 to 360 stays exactly the same.
Circle one: True False
Which is the expansion of sin(x + 45°)?
- sin x + sin 45°
- sin x + cos x + 45°
- cos x - sin x
- sin x cos 45° + cos x sin 45°
Maya collapses sin x cos 30° + cos x sin 30° back to sin(x + 30°) to solve. Is Maya right?
Circle one: True False
Sin of x plus 30 equals 1 with x between 0 and 360. What is x?
- 90 degrees
- 60 degrees
- 30 degrees
cos(x − 30°) = 0.5 for 0° ≤ x < 360°. What is the larger solution?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

30 · 45 − 30 = 15, so k = 30.
45° - 30° · 45 and 30 are both special with known sine and cosine. The 60 − 30 option makes 30°, not 15°.
Sin A cos B plus cos A sin B · Sine keeps the plus, with sin-cos plus cos-sin order.
False · Cosine flips the sign while sine keeps it.
0.5 · That is sin 30° = 0.5.
False · The shift rides along, so solving runs across 30 to 390.
sin x cos 45° + cos x sin 45° · Sine of a sum keeps the plus across products. Adding the sines directly is the top slip.
True · That is exactly the addition pattern.
60 degrees · The bracket must be 90, so subtract 30 to get 60.
330 · x − 30 = ±60 gives x = 90 or 330.