MathematicsChapter 6: Trigonometry and Bearing
Reciprocal Ratios
Each of sine, cosine, and tangent has a reciprocal partner.
Learning objectives
- Explain reciprocal ratios in clear mathematical language
- Apply the ideas in csc sec cot
- Solve problems involving when they fail
Definition
Each of sine, cosine, and tangent has a reciprocal partner. Cosecant is csc theta = 1/sin theta. Secant is sec theta = 1/cos theta. Cotangent is cot theta = 1/tan theta, which is also adj/opp or cos theta / sin theta. A reciprocal is defined only when the original ratio is not zero. These three names appear in identities and in some bearing problems with steep slopes.
Mathematical form
theta = 1/sin theta. Secant is sec theta
theta = 1/tan theta
theta = sin theta / cos theta and cot theta
Where / symbols
- csc theta = 1/sin theta
- Cosecant is csc theta = 1/sin theta.
- sec theta = 1/cos theta
- Secant is sec theta = 1/cos theta.
- cot theta = 1/tan theta
- Cotangent is cot theta = 1/tan theta, which is also adj/opp or cos theta / sin theta.
- undefined
- You cannot take 1/sin theta if sin theta = 0, so csc is undefined at 0 deg and 180 deg.
Relationship
- csc theta = 1/sin theta
- sec theta = 1/cos theta
- cot theta = 1/tan theta
Overview
Flip the original ratio, then watch zeros:
Csc Sec Cot
Write the three flips: csc theta = 1/sin theta, sec theta = 1/cos theta, and cot theta = 1/tan theta. In a right triangle that means csc = hyp/opp, sec = hyp/adj, and cot = adj/opp. If sin theta = 12 then csc theta = 2. If tan theta = 1 then cot theta = 1. Keep the names attached to the correct originals.
Examples sin=1/2 so csc=2 · cos=1/2 so sec=2 · tan=1 so cot=1
When They Fail
You cannot take 1/sin theta if sin theta = 0, so csc is undefined at 0 deg and 180 deg. Sec is undefined when cos theta = 0, such as 90 deg. Cotangent is cos theta / sin theta, so it is undefined when sin theta = 0; at 90 deg it equals 0. The shortcut cot theta = 1/tan theta is used only where tan theta is defined and nonzero. Always check the denominator before writing a reciprocal.
Examples csc undefined at 0 deg · sec undefined at 90 deg · cot undefined at 0 deg
What it means
Each of sine, cosine, and tangent has a reciprocal partner.
Main ideas
1. CSC SEC COT: Write the three flips: csc theta = 1/sin theta, sec theta = 1/cos theta, and cot theta = 1/tan theta. 2. WHEN THEY FAIL: You cannot take 1/sin theta if sin theta = 0, so csc is undefined at 0 deg and 180 deg.
Why it matters
A compact set: sin, cos, tan and their flips csc, sec, cot. Also tan theta = sin theta / cos theta and cot theta = cos theta / sin theta. Reciprocal is not the same as complementary: cos is not 1/sin.
How to answer in an exam
csc is 1/sin, not 1/cos. The names confuse people: sec pairs with cos, csc pairs with sin. If sin theta = 0 you must not write csc theta at all.
Explanation
A compact set: sin, cos, tan and their flips csc, sec, cot. Also tan theta = sin theta / cos theta and cot theta = cos theta / sin theta. Reciprocal is not the same as complementary: cos is not 1/sin.
In simple words
csc is 1/sin, not 1/cos. The names confuse people: sec pairs with cos, csc pairs with sin. If sin theta = 0 you must not write csc theta at all.
Key points
- csc theta = 1/sin theta
- sec theta = 1/cos theta
- cot theta = 1/tan theta
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Reciprocal Ratios
Sticky Notes · Quick Revision
Core Idea
- Each of sine, cosine, and tangent has a reciprocal partner.
Key Terms
- csc theta = 1/sin theta: Cosecant is csc theta = 1/sin theta.
- sec theta = 1/cos theta: Secant is sec theta = 1/cos theta.
Main Pattern
- 1. CSC SEC COT: Write the three flips: csc theta = 1/sin theta, sec theta = 1/cos theta, and cot theta = 1/tan theta.
- 2. WHEN THEY FAIL: You cannot take 1/sin theta if sin theta = 0, so csc is undefined at 0 deg and 180 deg.
Worked Reminder
- sin=12 so csc=2
- cos=12 so sec=2
- tan=1 so cot=1
Common Mistake
- csc is 1/sin, not 1/cos. The names confuse people: sec pairs with cos, csc pairs with sin. If sin theta = 0 you must not write csc theta at all.
Exam Tip
- State the rule for reciprocal ratios, show each algebraic step, and check the final answer against the question.
60-second revision
- Each of sine, cosine, and tangent has a reciprocal partner.
- Write the three flips: csc theta = 1/sin theta, sec theta = 1/cos theta, and cot theta = 1/tan theta.
- You cannot take 1/sin theta if sin theta = 0, so csc is undefined at 0 deg and 180 deg.
- A compact set: sin, cos, tan and their flips csc, sec, cot.
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