Complementary Angle Trig Relationships
Sine and cosine are linked through complementary angles: $\sin\theta = \cos(90° - \theta)$.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
The sine of an angle equals the cosine of its complement (and vice versa). It's cosine on the other side, not sine, and it only applies to angles that add to $90°$.
How to solve one, step by step
Example: $\sin 30°$ equals what cosine?
- The complement of $30°$ is $60°$.
- So $\sin 30° = \cos 60°$ (both $0.5$).
The mistakes that cost points
- Using sine on both sides. It's $\sin\theta = \cos(90° - \theta)$ — the function switches.
- Applying it to non-complementary angles. The identity needs the angles to sum to $90°$.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
If $\cos(k°) = \sin(25°)$, what is the value of $k$?
- A$25$
- B$115$
- C$65$
- D$155$
Show solution
Answer: C, $65$. $\sin\theta = \cos(90°-\theta)$, so $\sin(25°) = \cos(65°)$. Thus $k = 65$.
Medium
If $\sin(3k°) = \cos(k°)$, what is the value of $k$?
- A$22.5$
- B$30$
- C$45$
- D$90$
Show solution
Answer: A, $22.5$. $\sin(3k°) = \cos(90° - 3k°)$. Setting equal to $\cos(k°)$: $90° - 3k = k$, giving $90 = 4k$ and $k = 22.5$.
Hard
In a right triangle with acute angles $(4k)°$ and $(k + 25)°$, we know $\sin(4k°) = \cos(k + 25)°$. What is the value of $k$?
- A$13$
- B$31$
- C$18$
- D$\frac{90}{5}=18$
Show solution
Answer: A, $13$. $(4k) + (k+25) = 90$. $5k + 25 = 90$, giving $5k = 65$ and $k = 13$. Check: angles $52°$ and $38°$ — these sum to $90°$. ✓ And $\sin(52°) = \cos(38°)$. ✓
Find your exact gaps
Take a free, full-length practice test and see precisely which question types trip you up.
Practice this for freeNo credit card. Register once, take as many tests as you like.