Mastering SAT Trigonometry Basics: SOHCAHTOA and Beyond
Introduction SOHCAHTOA
Trigonometry is a small but important part of the SAT Math section. Questions typically involve sine (sin), cosine (cos), tangent (tan), and special right triangles. These concepts might seem challenging, but with simple strategies, you can master them.
In this guide, we’ll cover:
- Essential trigonometry concepts.
- Step-by-step problem-solving strategies.
- Common trigonometric mistakes and tips.
SOHCAHTOA: The Key to Right Triangles
SOHCAHTOA is a mnemonic that helps you remember trigonometric ratios:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
Example 1: Solving for Sine, Cosine, and Tangent
A right triangle has:
- Opposite = 3, Adjacent = 4, Hypotenuse = 5.
Find \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \).
Solution:
\[
\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}, \quad \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}, \quad \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4}
\]
Special Right Triangles
Memorise the two common special right triangles:
45°-45°-90° Triangle
- The legs are equal, and the hypotenuse is \( \text{leg} \times \sqrt{2} \).
Example:
If one leg = 5, the hypotenuse = \( 5\sqrt{2} \).
30°-60°-90° Triangle
- The sides follow the ratio \( x : x\sqrt{3} : 2x \).
Example:
If the shortest leg = 6:
- Long leg = \( 6\sqrt{3} \), Hypotenuse = 12.
Finding Missing Angles with Inverse Trigonometry
To find an angle:
- Use \( \sin^{-1} \), \( \cos^{-1} \), or \( \tan^{-1} \)** on your calculator.
Example:
If \( \sin \theta = 0.6 \), find \( \theta \).
Solution:
\[
\theta = \sin^{-1}(0.6) \approx 37^\circ
\]
SAT Trigonometry Practice Question
Question: In a right triangle, the opposite side = 8, the adjacent side = 6. Find \( \tan \theta \).
Solution:
\[
\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{8}{6} = \frac{4}{3}
\]
Common Trigonometry Mistakes
- Forgetting to label the sides of the triangle correctly.
- Mixing up sine, cosine, and tangent ratios.
- Misapplying the special triangle formulas.
Summary
Mastering trigonometry for the SAT involves understanding SOHCAHTOA, special triangles, and basic problem-solving strategies. With practice, you can tackle these questions confidently.
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