Combine a reference triangle with the signs in the stated quadrant.
Notes and key methods
Reference-angle magnitudes come from a right triangle; signs come from the quadrant. Sine is positive in quadrants I and II, cosine in I and IV, and tangent in I and III.
An angle outside one full revolution has the same sine and cosine as its coterminal angle. A negative angle reverses the sine sign while leaving cosine unchanged. State whether an angle is in degrees or radians.
Check the conditions
The quadrant is specified.
The reference angle is acute.
Three worked examples
Example 1 · Quadrants and general-angle ratios
θ is in quadrant 3. Its acute reference triangle has opposite side 32, adjacent side 60 and hypotenuse 68.
Find sin θ.
Find cos θ.
Find tan θ.
Find sin(−θ).
Show the reasoning
(a)
Use the reference triangle and quadrant sign.
\(\displaystyle - \frac{8}{17}\)
Answer: \(\displaystyle - \frac{8}{17}\).
(b)
Cosine is negative in the stated quadrant.
\(\displaystyle - \frac{15}{17}\)
Answer: \(\displaystyle - \frac{15}{17}\).
(c)
Divide the signed sine by the signed cosine.
\(\displaystyle \frac{8}{15}\)
Answer: \(\displaystyle \frac{8}{15}\).
(d)
Use the odd symmetry of sine.
\(\displaystyle \frac{8}{17}\)
Answer: \(\displaystyle \frac{8}{17}\).
Example 2 · Quadrants and general-angle ratios
θ is in quadrant 2. Its acute reference triangle has opposite side 24, adjacent side 7 and hypotenuse 25.
Find sin θ.
Find cos θ.
Find tan θ.
Find sin(−θ).
Show the reasoning
(a)
Use the reference triangle and quadrant sign.
\(\displaystyle \frac{24}{25}\)
Answer: \(\displaystyle \frac{24}{25}\).
(b)
Cosine is negative in the stated quadrant.
\(\displaystyle - \frac{7}{25}\)
Answer: \(\displaystyle - \frac{7}{25}\).
(c)
Divide the signed sine by the signed cosine.
\(\displaystyle - \frac{24}{7}\)
Answer: \(\displaystyle - \frac{24}{7}\).
(d)
Use the odd symmetry of sine.
\(\displaystyle - \frac{24}{25}\)
Answer: \(\displaystyle - \frac{24}{25}\).
Example 3 · Quadrants and general-angle ratios
θ is in quadrant 3. Its acute reference triangle has opposite side 10, adjacent side 24 and hypotenuse 26.
Find sin θ.
Find cos θ.
Find tan θ.
Find sin(−θ).
Show the reasoning
(a)
Use the reference triangle and quadrant sign.
\(\displaystyle - \frac{5}{13}\)
Answer: \(\displaystyle - \frac{5}{13}\).
(b)
Cosine is negative in the stated quadrant.
\(\displaystyle - \frac{12}{13}\)
Answer: \(\displaystyle - \frac{12}{13}\).
(c)
Divide the signed sine by the signed cosine.
\(\displaystyle \frac{5}{12}\)
Answer: \(\displaystyle \frac{5}{12}\).
(d)
Use the odd symmetry of sine.
\(\displaystyle \frac{5}{13}\)
Answer: \(\displaystyle \frac{5}{13}\).
Common mistakes
A square root alone does not determine the sign of a ratio.
Tangent signs are determined by both sine and cosine.