MathematicsChapter 6: Trigonometry and Bearing
Sine Cosine Tangent
On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta).
Learning objectives
- Explain sine cosine tangent in clear mathematical language
- Apply the ideas in unit circle
- Solve problems involving right triangle
Definition
On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta). In a right triangle, sin theta = opp/hyp, cos theta = adj/hyp, and tan theta = opp/adj. The three triangle ratios match the unit-circle idea because the hypotenuse plays the role of radius. Also tan theta = sin theta / cos theta when cos theta is not zero.
Mathematical form
theta = opp/hyp
theta = adj/hyp
theta = opp/adj. The three triangle ratios match
theta = sin theta / cos theta when cos theta is
Where / symbols
- (cos theta, sin theta)
- On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta).
- sin theta = opp/hyp
- In a right triangle, sin theta = opp/hyp, cos theta = adj/hyp, and tan theta = opp/adj.
- cos theta = adj/hyp
- In a right triangle, sin theta = opp/hyp, cos theta = adj/hyp, and tan theta = opp/adj.
- tan theta = opp/adj
- In a right triangle, sin theta = opp/hyp, cos theta = adj/hyp, and tan theta = opp/adj.
Relationship
- sin theta = opp/hyp
- cos theta = adj/hyp
- tan theta = opp/adj
Overview
Circle for any angle; triangle for an acute angle:
Unit Circle
Draw a circle of radius 1 centred at the origin. From the positive x-axis, mark angle theta. The landing point is (cos theta, sin theta): the x-coordinate is cosine and the y-coordinate is sine. This still works when theta is obtuse or a full turn, where a right triangle inside the first quadrant is no longer enough on its own.
Examples theta=0: (1, 0) · theta=90 deg: (0, 1) · theta=180 deg: (-1, 0)
Right Triangle
In a right triangle, the hypotenuse is opposite the right angle. For an acute angle theta, opposite is the side across from theta and adjacent is the other short side touching theta. Then sin theta = opp/hyp, cos theta = adj/hyp, and tan theta = opp/adj. Divide sine by cosine and the hyp cancels, so tan theta = sin theta / cos theta.
Examples opp=3, hyp=5, sin theta=3/5 · adj=4, hyp=5, cos theta=4/5 · opp=3, adj=4, tan theta=3/4
What it means
On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta).
Main ideas
1. UNIT CIRCLE: Draw a circle of radius 1 centred at the origin. 2. RIGHT TRIANGLE: In a right triangle, the hypotenuse is opposite the right angle.
Why it matters
SOH-CAH-TOA is only a memory hook: sine opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. On the unit circle the hypotenuse is 1, so sine is just the y-length and cosine is just the x-length.
How to answer in an exam
Do not swap opposite and adjacent. Adjacent touches the angle (but is not the hypotenuse). tan theta is undefined when cos theta = 0, for example at 90 deg, because you would divide by zero.
Explanation
SOH-CAH-TOA is only a memory hook: sine opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. On the unit circle the hypotenuse is 1, so sine is just the y-length and cosine is just the x-length.
In simple words
Do not swap opposite and adjacent. Adjacent touches the angle (but is not the hypotenuse). tan theta is undefined when cos theta = 0, for example at 90 deg, because you would divide by zero.
Key points
- (cos theta, sin theta)
- sin theta = opp/hyp
- cos theta = adj/hyp
- tan theta = opp/adj
- radius 1
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Sine Cosine Tangent
Sticky Notes · Quick Revision
Core Idea
- On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta).
Key Terms
- (cos theta, sin theta): On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta).
- sin theta = opp/hyp: In a right triangle, sin theta = opp/hyp, cos theta = adj/hyp, and tan theta = opp/adj.
Main Pattern
- 1. UNIT CIRCLE: Draw a circle of radius 1 centred at the origin.
- 2. RIGHT TRIANGLE: In a right triangle, the hypotenuse is opposite the right angle.
Worked Reminder
- theta=0: (1, 0)
- theta=90 deg: (0, 1)
- theta=180 deg: (-1, 0)
Common Mistake
- Do not swap opposite and adjacent. Adjacent touches the angle (but is not the hypotenuse). tan theta is undefined when cos theta = 0, for example at 90 deg, because you would divide by zero.
Exam Tip
- State the rule for sine cosine tangent, show each algebraic step, and check the final answer against the question.
60-second revision
- On the unit circle (radius 1) a point at angle theta in standard position has coordinates (cos theta, sin theta).
- Draw a circle of radius 1 centred at the origin.
- In a right triangle, the hypotenuse is opposite the right angle.
- SOH-CAH-TOA is only a memory hook: sine opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent.
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