The secant of 30 degrees, denoted as sec(30°), is a trigonometric function that represents the ratio of the length of the hypotenuse to the length of the adjacent side in a right triangle. In the context of the unit circle, sec(30°) is defined as the reciprocal of the cosine function, specifically $$sec(30° = \frac{1}{cos(30°)}$$. This connection illustrates how secant relates to the geometric representation of angles and circles.
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The value of sec(30°) can be calculated as $$sec(30° = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$$.
In decimal form, sec(30°) is approximately 1.1547.
Secant is undefined for angles where cosine is zero, but sec(30°) is defined since cos(30°) is not zero.
The secant function has a periodicity of 360 degrees, meaning sec(30°) = sec(390°) = sec(-330°).
Secant can be used to find other trigonometric functions through identities; for example, $$sec^2(θ) = 1 + tan^2(θ)$$.
Review Questions
How do you derive the value of sec(30°) from the unit circle?
To find sec(30°) using the unit circle, first identify the cosine of 30 degrees. The point on the unit circle corresponding to 30 degrees has coordinates $$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$. Since secant is the reciprocal of cosine, we calculate it as $$sec(30° = \frac{1}{cos(30°)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$$.
Compare and contrast sec(30°) with another trigonometric function like tan(30°). What are their relationships?
Sec(30°) and tan(30°) are both trigonometric functions but represent different ratios. While sec(30°) is defined as $$sec(30° = \frac{2}{\sqrt{3}}$$, tan(30°) equals $$\frac{1}{\sqrt{3}}$$. The relationship between them can be seen through identities; for instance, $$sec^2(θ) = 1 + tan^2(θ)$$ applies here. Plugging in tan(30°), we find that $$sec^2(30° = 1 + (\frac{1}{\sqrt{3}})^2 = 1 + \frac{1}{3} = \frac{4}{3}$$ which matches our earlier calculation.
Evaluate how understanding sec(30°) and its relationship with other functions influences problem-solving in trigonometry.
Understanding sec(30°) and its interrelations with functions like cosine and tangent enhances problem-solving skills by providing multiple pathways to find unknowns in trigonometric problems. For example, if you know sec(30°), you can easily derive related values like cos(30°) and tan(30°), allowing for a more holistic approach to complex problems. This interconnected knowledge helps in simplifying calculations and verifying solutions within different contexts.
Related terms
Cosine: The cosine function gives the ratio of the adjacent side to the hypotenuse in a right triangle.
A circle with a radius of one, centered at the origin, where each point on the circle corresponds to an angle and its coordinates represent trigonometric values.
Trigonometric Ratios: Ratios that relate the angles and sides of triangles, including sine, cosine, and tangent.