Trigonometric ratios (sin, cos, tan) for acute angles
Trigonometric Ratios
Trigonometric Ratios is a topic in Right Triangles & Trigonometry in the Common Core State Standards. It is aligned to Standard G-SRT.6, which requires students to understand that side ratios in right triangles are properties of the angles, defining the trigonometric ratios for acute angles.
The trigonometric ratios sine, cosine, and tangent are side ratios in a right triangle, remembered as SOH-CAH-TOA.
Theory
For an acute angle \(\theta\) in a right triangle, the sides are named relative to \(\theta\): the opposite, the adjacent, and the hypotenuse. The three trigonometric ratios are
remembered by SOH-CAH-TOA.
The three ratios and the inverse:
How to use the ratios
- Label the sides opposite, adjacent, hypotenuse for the angle.
- Choose the ratio linking the known and unknown (SOH-CAH-TOA).
- Solve for a side, or use the inverse function for an angle.
Sine is opposite over hypotenuse.
| \(\sin\theta\) | \(=\) | \(\dfrac{3}{5}\) |
Rearrange \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\).
| \(\text{opp}\) | \(=\) | \(10\sin 40^\circ\) |
| \(\approx\) | \(6.43\) |
Use \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\).
| \(\text{opp}\) | \(=\) | \(12\tan 35^\circ\) |
| \(\approx\) | \(8.40\) |
Use the inverse tangent.
| \(\theta\) | \(=\) | \(\tan^{-1}\!\dfrac{7}{24}\) |
| \(\approx\) | \(16.3^\circ\) |
Common pitfalls
Frequently asked questions
What does SOH-CAH-TOA mean?
Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
How do you find a missing side with trigonometry?
Choose the ratio relating the known side and angle to the unknown side, then solve.
How do you find an angle from two sides?
Use an inverse trig function, such as \(\theta=\tan^{-1}(\text{opp}/\text{adj})\).
Which side is the hypotenuse?
The longest side, opposite the right angle. It is the same regardless of which acute angle you use.