Reference

Trigonometric Ratios

Concept

Trigonometric Ratios

A trigonometric ratio relates two side lengths of a right triangle. Consider the right triangle $△ A B C .$ One of its acute angles has been named $θ .$

Since it is opposite to the right angle, $\overline{B C}$ is the hypotenuse of the right triangle. The remaining sides — the legs — can be named relative to the marked angle $θ .$ Because $\overline{A B}$ is next to $∠ θ,$ it is called the adjacent side. Conversely, because $\overline{A C}$ lies across from $∠ θ,$ it is called the opposite side.

Trigonometric ratios can be used to determine unknown side lengths or angle measures in right triangles.

Concept

Sine

For an acute angle

θ

of a right triangle, the sine of

θ

is the ratio between the lengths of the opposite side and the hypotenuse.

The sine of

θ

is written as

sin θ .

$sin θ = \frac{Opposite}{Hypotenuse}$

This trigonometric ratio states the ratio between the opposite side and the hypotenuse. It gives no indication about the lengths of the individual sides.

Right triangle with two vertices movable and the sine of one acute angle is computed while the measures change

It should be noted that for a given angle, when the opposite side and hypotenuse measurements change, if their ratio stays the same, then the sine, too, would stay the same.

Concept

Cosine

For an acute angle

θ

of a right triangle, the cosine of

θ

is the ratio between the lengths of the adjacent side and the hypotenuse.

The cosine of

θ

is written as

cos θ .

$cos θ = \frac{Adjacent}{Hypotenuse}$

While this trigonometric ratio expresses the ratio of the adjacent side to the hypotenuse, it does not state the actual measurements of their lengths.

Right triangle with two vertices movable and the cosine of one acute angle is computed while the measures change

It should be noted that for a given angle, when the adjacent side and hypotenuse lengths change, if their ratio stays the same, then the cosine, too, would stay the same.

Concept

Tangent

For an acute angle

θ

of a right triangle, the tangent of

θ

is the ratio between the lengths of the opposite side and the adjacent side.

The tangent of

θ

is written as

tan θ .

$tan θ = \frac{Opposite}{Adjacent}$

The trigonometric ratio expresses the ratio between the opposite side and the adjacent side. It does not give any indication about the actual measurement of the side lengths.

Right triangle with two vertices movable and the tangent of one acute angle is computed while the measures change

It should be noted that for a given angle, when the adjacent side and the opposite side lengths change, if their ratio stays the same, then the tangent, too, would stay the same.

Concept

Cosecant

For an acute angle

θ

of a right triangle, the cosecant of

θ

is the ratio between the lengths of the hypotenuse and the opposite side.

The cosecant of

θ

is written as

csc (θ) .

$csc (θ) = \frac{hypotenuse}{opposite}$

This trigonometric ratio states the ratio between the hypotenuse and the opposite side to a certain angle. It gives no indication about the lengths of the sides.

Right triangle with two vertices movable and the cosecant of one acute angle is computed while the measures change

When the opposite side and hypotenuse lengths change for a given angle, if their ratio stays the same then the cosecant, too, would stay the same. Also note that the cosecant of

θ

is the reciprocal of the sine of

θ .

⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ csc θ = \frac{hyp}{opp} \frac{∣}{∣} sin θ = \frac{opp}{hyp} \frac{\frac{∣}{∣}}{∣} \Rightarrow csc θ = \frac{1}{sin θ}

Concept

Secant

For an acute angle

θ

of a right triangle, the secant of

θ

is the ratio between the lengths of the hypotenuse and the adjacent side.

The secant of

θ

is written as

sec (θ) .

$sec (θ) = \frac{hypotenuse}{adjacent}$

This trigonometric ratio states the ratio between the hypotenuse and the adjacent side to a certain angle. It gives no indication about the lengths of the sides.

Right triangle with two vertices movable and the secant of one acute angle is computed while the measures change

When the adjacent side and hypotenuse lengths change for a given angle, if their ratio stays the same then the secant, too, would stay the same. Also note that the secant of

θ

is the reciprocal of the cosine of

θ .

⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ sec θ = \frac{hyp}{adj} \frac{∣}{∣} cos θ = \frac{adj}{hyp} \frac{\frac{∣}{∣}}{∣} \Rightarrow sec θ = \frac{1}{cos θ}

Concept

Cotangent

For an acute angle

θ

of a right triangle, the cotangent of

θ

is the ratio between the lengths of the adjacent side and the opposite side.

The cotangent of

θ

is written as

cot θ .

$cot θ = \frac{adjacent}{opposite}$

This trigonometric ratio states the ratio between the adjacent side to a certain angle and the opposite side to the angle. It gives no indication about the lengths of the sides.

Right triangle with two vertices movable and the cotangent of one acute angle is computed while the measures change

When the adjacent side and the opposite side lengths for a given angle change, if their ratio stays the same, then the cotangent, too, would stay the same. In addition, note that the cotangent of

θ

is also the reciprocal of the tangent of

θ .

⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ cot θ = \frac{adj}{opp} \frac{∣}{∣} tan θ = \frac{opp}{adj} \frac{\frac{∣}{∣}}{∣} \Rightarrow cot θ = \frac{1}{tan θ}

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