Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\tan 70^{\circ}-\frac{\sin 70^{\circ}}{\cos 70^{\circ}}$$

Short answer

0

Step-by-step solution

- Recall the definition of the tangent function

The tangent function is defined by \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{sin} \theta}{\text{cos} \theta}\).

- Substitute the given angle into the tangent definition

For \(\theta = 70^{\circ}\), we have \(\tan 70^{\circ} = \frac{\text{sin} 70^{\circ}}{\text{cos} 70^{\circ}}\).

- Write the original expression with the substitution

The given expression is \(\tan 70^{\circ}-\frac{\text{sin} 70^{\circ}}{\text{cos} 70^{\circ}}\). Substituting \(\tan 70^{\circ}\) gives: \(\frac{\text{sin} 70^{\circ}}{\text{cos} 70^{\circ}} - \frac{\text{sin} 70^{\circ}}{\text{cos} 70^{\circ}}\).

- Simplify the expression

Both terms in the expression are the same: \(\frac{\text{sin} 70^{\circ}}{\text{cos} 70^{\circ}} - \frac{\text{sin} 70^{\circ}}{\text{cos} 70^{\circ}} = 0\).

Key concepts

tangent function

The tangent function is one of the primary functions in trigonometry. It’s denoted as \( \tan \) and is defined as the ratio of the sine function to the cosine function of the same angle. Mathematically, we express it as \( \tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \). This means for any angle \( \theta \), you can get its tangent value by dividing the sine of the angle by its cosine. For example, for an angle of 70 degrees, the tangent is \( \tan 70^\text{\degree} = \frac{\text{sin} 70^\text{\degree}}{\text{cos} 70^\text{\degree}} \). The tangent function is very useful when dealing with right triangles and helps in converting between different trigonometric forms.

sine function

The sine function is another fundamental trigonometric function. It is denoted as \( \text{sin} \). The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. Symbolically, \( \text{sin} \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \). For instance, if we take \( \theta = 70^\text{\degree} \), the sine of this angle is \( \text{sin} 70^\text{\degree} \), which represents the ratio mentioned. The sine function ranges between -1 and 1 for any angle and is periodic with a period of \( 360^\text{\degree} \) or \( 2\text{\pi} \) radians.

cosine function

The cosine function is crucial in trigonometry. It’s denoted as \( \text{cos} \) and is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. So, \( \text{cos} \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \). For example, for an angle \( 70^\text{\degree} \), the cosine value is given by \( \text{cos} 70^\text{\degree} \). Like the sine function, the cosine function also ranges between -1 and 1 and has the same period of \( 360^\text{\degree} \) or \( 2\text{\pi} \) radians. It helps in determining the relationship between the angle and the lengths of the sides of a triangle.

trigonometric simplification

Trigonometric simplification involves using fundamental identities to reduce complex expressions to simpler forms. In our example, we started with \( \tan 70^\text{\degree} - \frac{\text{sin} 70^\text{\degree}}{\text{cos} 70^\text{\degree}} \). By substituting the identity \( \tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \), we rewrite it as \( \frac{\text{sin} 70^\text{\degree}}{\text{cos} 70^\text{\degree}} - \frac{\text{sin} 70^\text{\degree}}{\text{cos} 70^\text{\degree}} \). Simplifying further, we realize these terms cancel each other out, making the expression equal to 0. This is a perfect example of how understanding and applying basic trigonometric identities can simplify and solve expressions efficiently.