Let's consider a right triangle $A B C,$ and let's write the tangent ratio of one acute angle.

We are interested in finding the measure of $∠ A$ for which $tan A$ is equal to $1,$ greater than $1,$ and less than $1 .$ Let's study each case separately.

Equal to $1$

We begin this case by considering

tan A = 1

and substituting it into the tangent ratio written above.

\frac{B C}{A C} = tan A = 1 \Rightarrow A C = B C

From the above, we conclude that

△ A B C

is an isosceles triangle.

Since $△ A B C$ is an isosceles right triangle, we have that the measure of both acute angles is $4 5^{\circ} .$

The tangent of a $4 5^{\circ}$ angle equals $1 .$

Greater Than $1$

Let's suppose that

tan A > 1

and substitute it into the tangent ratio equation.

\frac{B C}{A C} = tan A > 1 \Rightarrow B C > A C

From the latter inequality and the Triangle Longer Side Theorem, we conclude that

m ∠ A > m ∠ B .

Now, we will add

m ∠ A

to both sides of the inequality written above and use the fact that

∠ A

and

∠ B

are complementary.

m ∠ A > m ∠ B

AddIneq

$LHS + m ∠ A > RHS + m ∠ A$

m ∠ A + m ∠ A > m ∠ A + m ∠ B

▼

Solve for

m ∠ A

AddTerms

Add terms

2 m ∠ A > m ∠ A + m ∠ B

Substitute

$m ∠ A + m ∠ B = 9 0^{\circ}$

2 m ∠ A > 9 0^{\circ}

DivIneq

$LHS / 2 > RHS / 2$

m ∠ A > 4 5^{\circ}

That way, we've found the angle measures for which the tangent is greater than

1 .

The tangent of an acute angle whose measure is greater than $4 5^{\circ}$ is greater than $1 .$

Less Than $1$

Finally, let's suppose that

tan A < 1

and substitute it into the equation written at the beginning.

\frac{B C}{A C} = tan A < 1 \Rightarrow B C < A C

Again, from the latter inequality and the Triangle Longer Side Theorem, we conclude that

m ∠ A < m ∠ B .

As before, we will add

m ∠ A

to both sides of the inequality above and use the fact that

∠ A

and

∠ B

are complementary.

m ∠ A < m ∠ B

AddIneq

$LHS + m ∠ A < RHS + m ∠ A$

m ∠ A + m ∠ A < m ∠ A + m ∠ B

▼

Solve for

m ∠ A

AddTerms

Add terms

2 m ∠ A < m ∠ A + m ∠ B

Substitute

$m ∠ A + m ∠ B = 9 0^{\circ}$

2 m ∠ A < 9 0^{\circ}

DivIneq

$LHS / 2 < RHS / 2$

m ∠ A < 4 5^{\circ}

From the latter inequality, we can write the following conclusion.

The tangent of an acute angle whose measure is less than $4 5^{\circ}$ is less than $1 .$

Equal to 1

Greater Than 1

Less Than 1

Equal to $1$

Greater Than $1$

Less Than $1$

Exercise

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