Basic Trigonometry

1Summary 重點摘要

Learn sine, cosine, tangent, use special angles 30° 45° 60°.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Know the right triangleA right triangle has one interior angle of exactly 90°, and its longest side is called the hypotenuse, always opposite the right angle. The names of the other two sides depend on which acute angle θ we choose: the side facing θ is the opposite side, and the side next to θ that is not the hypotenuse is the adjacent side. A common beginner's mistake is forgetting that 'opposite' and 'adjacent' change when a different angle is chosen, so always mark θ before naming the sides.
2. Step 2: The SOHCAHTOA mnemonicThere are three basic ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. The mnemonic SOHCAHTOA links the initials of these definitions to help you remember them. Note that all three are ratios of two sides, so knowing any two sides, or one side and one angle, is enough to build an equation. The usual error is mixing up opposite with adjacent and using sin where cos is needed, so label all three sides before you start.
3. Step 3: Identify opposite and adjacent sidesBefore solving, mark the angle θ on the diagram and name all three sides: the hypotenuse always faces the right angle and never changes; the opposite side lies directly across from θ; the adjacent side touches θ and is not the hypotenuse. If the question asks about the other acute angle instead, the opposite and adjacent sides swap, which is the most common source of lost marks. Lightly writing the letters H, O and A beside the sides, then checking once more before substituting, greatly reduces errors.
4. Step 4: Choose the suitable ratioChoose the ratio by observing which two sides are known and which side is required: use sin when the hypotenuse and opposite are involved, cos when the hypotenuse and adjacent are involved, and tan when the opposite and adjacent are involved. The principle is that the equation may contain only one unknown; if two unknown sides appear in the formula, the wrong ratio has been chosen. When the angle is unknown but two sides are given, pick the ratio the same way and use an inverse function, for example θ = sin⁻¹(opposite/hypotenuse).
5. Step 5: Learn the special angle valuesMemorise the values for 30°, 45° and 60°: sin 30° = 12, cos 30° = √32, tan 30° = 1/√3; sin 45° = cos 45° = √22, tan 45° = 1; sin 60° = √32, cos 60° = 12, tan 60° = √3. To remember them, derive them yourself from the 30-60-90 and 45-45-90 special right triangles, so that even a forgotten value can be rebuilt. Examinations often require surd form, so do not convert to decimals too early, or accuracy will be lost.
6. Step 6: Set up an equation to find an unknown sideTo find an unknown side, first choose the suitable ratio, write an equation such as sin 30° = opposite/hypotenuse, substitute the known values, and finally rearrange to solve. Keep plenty of decimal places throughout the working, usually four, or store intermediate values in the calculator's memory, and round only at the last step to the accuracy the question demands. A common error is rounding too early, which shifts the final answer outside the accepted range and costs marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Right triangle first can use trig ratio
• SOHCAHTOA: Sin=opp/hyp, Cos=adj/hyp, Tan=opp/adj
• Hypotenuse longest, opposite right angle
• Special angles: 30° 45° 60° must remember values
• See known unknown sides, choose which trig ratio