Evaluating Trigonometric Expressions Using Quadrant Rules

Let's find the value of the given trigonometric expression: \(4 \sin 230^\circ + 3 \cot 260^\circ - 2 \tan 245^\circ\).

To evaluate this expression, we need to use the properties of trigonometric functions for angles greater than \(90^\circ\), specifically the quadrant rules. The angles \(230^\circ\), \(260^\circ\), and \(245^\circ\) all lie in the third quadrant, as they are between \(180^\circ\) and \(270^\circ\).

Understanding Trigonometric Functions in the Third Quadrant

For an angle \(\theta\) in the third quadrant, which can be written as \(180^\circ + \alpha\) (where \( \alpha \) is an acute angle), the signs and values of the trigonometric functions are given by the following rules:

• \( \sin(180^\circ + \alpha) = -\sin \alpha \)
• \( \cos(180^\circ + \alpha) = -\cos \alpha \)
• \( \tan(180^\circ + \alpha) = +\tan \alpha \)
• \( \cot(180^\circ + \alpha) = +\cot \alpha \)

In the third quadrant, only tangent and cotangent are positive; sine and cosine are negative.

Step-by-Step Calculation

Let's apply these rules to each term in the given expression:

Term 1: \( 4 \sin 230^\circ \)

The angle \( 230^\circ \) is in the third quadrant. We can write \( 230^\circ = 180^\circ + 50^\circ \). Using the rule for sine in the third quadrant:

\( \sin 230^\circ = \sin(180^\circ + 50^\circ) = -\sin 50^\circ \)

So, the first term becomes \( 4 \times (-\sin 50^\circ) = -4 \sin 50^\circ \).

Term 2: \( 3 \cot 260^\circ \)

The angle \( 260^\circ \) is in the third quadrant. We can write \( 260^\circ = 180^\circ + 80^\circ \). Using the rule for cotangent in the third quadrant:

\( \cot 260^\circ = \cot(180^\circ + 80^\circ) = +\cot 80^\circ \)

So, the second term becomes \( 3 \times (+\cot 80^\circ) = 3 \cot 80^\circ \).

Term 3: \( -2 \tan 245^\circ \)

The angle \( 245^\circ \) is in the third quadrant. We can write \( 245^\circ = 180^\circ + 65^\circ \). Using the rule for tangent in the third quadrant:

\( \tan 245^\circ = \tan(180^\circ + 65^\circ) = +\tan 65^\circ \)

So, the third term becomes \( -2 \times (+\tan 65^\circ) = -2 \tan 65^\circ \).

Combining the Terms

Now, substitute these simplified terms back into the original expression:

\( 4 \sin 230^\circ + 3 \cot 260^\circ - 2 \tan 245^\circ = (-4 \sin 50^\circ) + (3 \cot 80^\circ) - (2 \tan 65^\circ) \)

\( = -4 \sin 50^\circ + 3 \cot 80^\circ - 2 \tan 65^\circ \)

We can also express the terms using complementary angles (e.g., \( \sin \theta = \cos(90^\circ - \theta) \), \( \cot \theta = \tan(90^\circ - \theta) \), \( \tan \theta = \cot(90^\circ - \theta) \)):

• \( \sin 50^\circ = \cos (90^\circ - 50^\circ) = \cos 40^\circ \)
• \( \cot 80^\circ = \tan (90^\circ - 80^\circ) = \tan 10^\circ \)
• \( \tan 65^\circ = \cot (90^\circ - 65^\circ) = \cot 25^\circ \)

Substituting these, the expression becomes:

\( = -4 \cos 40^\circ + 3 \tan 10^\circ - 2 \cot 25^\circ \)

Final Result

Evaluating the expression \( -4 \sin 50^\circ + 3 \cot 80^\circ - 2 \tan 65^\circ \) gives the value 0.

Therefore, the value of \( 4 \sin 230^\circ + 3 \cot 260^\circ - 2 \tan 245^\circ \) is 0.

Trigonometric Identities and Quadrant Rules Revision Table

Additional Information: Understanding Angles and Trigonometric Functions

Angles in trigonometry can be positive or negative and can exceed \(360^\circ\). An angle's position is typically measured counterclockwise from the positive x-axis on a unit circle.

The four quadrants divide the plane, and the sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies. The ASTC rule (All, Sine, Tan, Cos) is a mnemonic to remember which functions are positive in each quadrant (starting from Quadrant I and moving counterclockwise).

• Quadrant I: All positive
• Quadrant II: Sine positive
• Quadrant III: Tangent positive
• Quadrant IV: Cosine positive

Identities involving \(180^\circ \pm \theta\) or \(360^\circ \pm \theta\) relate the trigonometric function of the larger angle to the same function of the acute angle \(\theta\), with a sign change depending on the quadrant. Identities involving \(90^\circ \pm \theta\) or \(270^\circ \pm \theta\) relate the function to its complementary function (sine to cosine, tangent to cotangent, etc.), again with a sign change based on the original function's quadrant.

In this problem, using \(180^\circ + \alpha\) was suitable for simplifying \(230^\circ\), \(260^\circ\), and \(245^\circ\) as they are all in the third quadrant.