# Conversion Relations of Trigonometric Ratios — Table

TL;DR

Conversion relations let you write any one of the six trigonometric ratios in terms of any other — for example, expressing sin⁡θ, sec⁡θ, and tan⁡θ all in terms of cot⁡θ. The method chains three engines: the reciprocal relations, the quotient relations, and the Pythagorean identities. This article gives the full conversion table, the step-by-step method, the sign caveat, and six worked examples — including the standard textbook questions.

## What Are the Conversion Relations of Trigonometric Ratios?

**The conversion relations of trigonometric ratios are the rules for expressing any of the six ratios in terms of any other** — sine in terms of cosine, all six in terms of tan⁡θ, and so on. The point is _interconversion_: starting from one known ratio and rebuilding the rest, without going back to the triangle's side lengths.

A quick note on the name, because the phrase is used two ways. Some sources use "conversion relations" for _angle transformations_ — turning sin⁡(90°+θ) into cos⁡θ, for instance. This article is about the other, more common classroom meaning: **expressing one ratio through another**. The angle-transformation idea is covered separately in [trigonometric ratios of complementary angles](/content/math/trigonometry/trigonometric-ratios-of-complementary-angles/index.html).

The conversions are powered by the [basic properties of trigonometric ratios](/content/math/trigonometry/basic-properties-of-trigonometric-ratios/index.html), grouped into three engines.

- **Reciprocal relations** — csc⁡θ=1/sin⁡θ, sec⁡θ=1/cos⁡θ, cot⁡θ=1/tan⁡θ.

- **Quotient relations** — tan⁡θ=sin⁡θ/cos⁡θ, cot⁡θ=cos⁡θ/sin⁡θ.

- **Pythagorean identities** — sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=csc²θ.

## The Full Conversion Table

Here is the complete reference: each of the six ratios written in terms of each base ratio. The table assumes θ is acute (Quadrant I), so every value is positive; the sign caveat for other quadrants comes after.

| In terms of → | sin⁡θ | cos⁡θ | tan⁡θ |
| --- | --- | --- | --- |
| sin⁡θ | sin⁡θ | 1−cos²θ | tan⁡θ/ |
| cos⁡θ | 1−sin²θ | cos⁡θ | 1/tan²θ |
| tan⁡θ | sin⁡θ/1−sin²θ | 1/cos²θ | tan⁡θ |
| csc⁡θ | 1/sin⁡θ | 1/cos²θ | 1+tan²θ/tan⁡θ |
| sec⁡θ | 1/sin²θ | 1/cos⁡θ | 1+tan²θ |
| cot⁡θ | 1−sin²θ/sin⁡θ | cos⁡θ/(1−cos²θ) | 1/tan⁡θ |

And the same six ratios in terms of the reciprocal base ratios:

| In terms of → | csc⁡θ | sec⁡θ | cot⁡θ |
| --- | --- | --- | --- |
| sin⁡θ | 1/csc⁡θ | sec²θ−1/sec⁡θ | 1+cot²θ |
| cos⁡θ | csc²θ−1/csc⁡θ | 1/sec⁡θ | cotθ/1+cot²θ |
| tan⁡θ | 1/csc²θ−1 | sec²θ−1 | 1/cot⁡θ |
| sec⁡θ | cscθ/csc²θ−1 | sec⁡θ | 1+cot²θ |
| csc⁡θ | csc⁡θ | sec²θ−1/sec²θ | 1+cot²θ |
| cot⁡θ | 1/csc²θ−1 | 1/sec²θ−1 | cot⁡θ |

## How Do You Convert One Trigonometric Ratio Into Another?

The method is a fixed three-step chain. Suppose you are given sin⁡θ and want everything else.

1. **Get cosine from the Pythagorean identity.** Since sin²θ+cos²θ=1:
   
   cos⁡θ=√(1−sin²θ)

2. **Get tangent from the quotient relation.** Now that both sine and cosine are known:
   
   tan⁡θ=sin⁡θ/cos⁡θ=sin⁡θ/√(1−sin²θ)

3. **Get the reciprocals by flipping.** Each of csc⁡θ, sec⁡θ, cot⁡θ is 1 over the matching primary ratio.

csc⁡θ=1/sin⁡θ, sec⁡θ=1/√(1−sin²θ), cot⁡θ=√(1−sin²θ)/sin⁡θ.

The same three steps work from any starting ratio. If you start from tan⁡θ, use 1+tan²θ=sec²θ to get secant first; if you start from cot⁡θ, use 1+cot²θ=csc²θ. The Pythagorean identity always supplies the "missing partner," and the quotient and reciprocal relations finish the job.

### What about the sign?

The Pythagorean step produces a square root, which carries a ±. For acute angles the sign is always positive. For angles beyond 90°, the sign is fixed by the quadrant the angle lands in — the ASTC rule from the basic properties decides whether to take the + or the −. Drop the sign check and a Quadrant II answer comes out wrong.

## Examples of Conversion Relations of Trigonometric Ratios

### Example 1

**Express cos⁡θ in terms of sin⁡θ (acute angle).**

From the Pythagorean identity sin²θ+cos²θ=1:

cos²θ=1−sin²θ

cos⁡θ=√(1−sin²θ)

**Final answer:** cos⁡θ=√(1−sin²θ) (positive, since θ is acute).

### Example 2

**Express sin⁡θ in terms of tan⁡θ.**

The first instinct is to write sin⁡θ=tan⁡θ·cos⁡θ and stop, but that still contains cos⁡θ, so it is not yet "in terms of tan⁡θ." The correct route uses 1+tan²θ=sec²θ:

cos⁡θ=1/sec⁡θ=1/√(1+tan²θ);

then:

sin⁡θ=tan⁡θ·cos⁡θ=tan⁡θ·1/√(1+tan²θ).

**Final answer:** sin⁡θ=tan⁡θ/√(1+tan²θ) — now entirely in terms of tan⁡θ.

### Example 3

**Express the trigonometric ratios sin⁡A, sec⁡A, and tan⁡A in terms of cot⁡A.**

Start from cot⁡A and use 1+cot²A=csc²A:

csc⁡A=√(1+cot²A);

since sin⁡A=1/csc⁡A:

sin⁡A=1/√(1+cot²A);

for tan⁡A, use the reciprocal relation directly:

tan⁡A=1/cot⁡A;

for sec⁡A, get cosine from
cos⁡A=cot⁡A·sin⁡A=cot⁡A/√(1+cot²A), then flip:

sec⁡A=1/cos⁡A=√(1+cot²A)/cot⁡A.

**Final answer:** sin⁡A=1/√(1+cot²A), tan⁡A=1/cot⁡A, sec⁡A=√(1+cot²A)/cot⁡A.

### Example 4

**Write all the other trigonometric ratios of ∠A in terms of sec⁡A.**

Use 1+tan²A=sec²A, so tan⁡A=√(sec²A−1);

cos⁡A=1/sec⁡A;

sin⁡A=tan⁡A·cos⁡A=√(sec²A−1)/sec⁡A;

csc⁡A=1/sin⁡A=sec⁡A/√(sec²A−1);

cot⁡A=1/tan⁡A=1/√(sec²A−1).

**Final answer:** the five ratios as written above, each expressed purely in sec⁡A.

### Example 5

**Given tan⁡θ=34 for an acute angle, find sin⁡θ and cos⁡θ using conversion.**

From 1+tan²θ=sec²θ:

sec²θ=1 + 916=2516 ⟹ sec⁡θ=54;

cos⁡θ=1/sec⁡θ=45;

sin⁡θ=tan⁡θ·cos⁡θ=34·45=35.

**Final answer:** sin⁡θ=35, cos⁡θ=45 — the familiar 333-444-555 triangle, recovered from tangent alone.

### Example 6

**An angle θ in Quadrant II has sin⁡θ=513. Convert to find cos⁡θ and tan⁡θ.**

The conversion gives the magnitude:

cos⁡θ=±√(1−25/169)=±12/13;

now apply the sign caveat. In Quadrant II, cosine is negative, so:

cos⁡θ=−12/13, tan⁡θ=sin⁡θ/cos⁡θ=5/13−12/13=−5/12.

**Final answer:** cos⁡θ=−12/13, tan⁡θ=−5/12. The conversion supplies the size; the quadrant fixes the sign.

## Why Conversion Is the Skill, Not the Table

The conversions matter because they turn one piece of information into all of it — and because the _method_ is reusable in a way the table is not.

- **They solve "given one ratio, find another" instantly.** This is one of the most common question shapes in introductory courses and beyond, and the three-engine chain answers every version of it.

- **They are the backbone of identity proofs.** To prove an identity, you almost always rewrite everything in terms of sin and cos — which _is_ a conversion. The wider toolkit lives in [trigonometric identities](/content/math/trigonometry/trigonometric-identities/index.html).

- **They scale to any quadrant.** Once the acute-angle conversions are automatic, extending them is just a sign decision from ASTC — which opens the door to the unit circle and [trigonometric functions](/content/math/trigonometry/trigonometric-functions/index.html) of any angle.