The word TRIGONOMETRY is Greek; tri means three, gon means angle and meter means to measure. Combining all these three words, we get, “three angle measure”. The logic behind this is that trigonometry involves the measurements inside a triangle, which has three angles.
Trigonometry is based upon three primary relationships within a triangle. The mnemonic SOHCAHTOA has been commonly used to help students memorize the associations between the sine, cosine, and tangent in a right triangle. These trigonometric identities are used to find missing components of a right triangle.
SOH: Sine of θ equals the Opposite side divided by the Hypotenuse
\(sin θ =\frac{Opp}{Hyp}\)
CAH: Cosine of θ equals Adjacent side divided by the Hypotenuse
\(cos θ =\frac{Adj}{Hyp}\)
TOA: Tangent of θ equals Opposite side divided by the Adjacent
\(tan θ =\frac{Opp}{Adj}\)
The Sine, cosine and tangent of some common angles are shown in the table below:
Trigonometry ratios can be expressed in terms of each other as well. The ratio of the sine of θ to the cosine of θ is equal to the tangent of θ:\(tan θ =\frac{sin θ }{cos θ }\)
You may also encounter some inverse trigonometric ratios in an SAT exam.
Solved Example 2
In the xy-plane, O is the center of the circle, and the measure of ∠AOB is \(\frac{\pi }{a}\) radians. What is the value of a?
A. 4
B. 6
C. 8
D. 12
Solution: By the distance formula, the length of radius OA is \(\sqrt{(\sqrt{3})^{2}+1^{2}} = 2\)
Thus Sin(∠AOB) = 1/2 .
Therefore, ∠AOB=30° , which equals to \(30\frac{\left ( \pi \right )}{180}=\frac{\pi }{6}radians\)
Hence, the value of a is 6. The correct answer is B.
We know that in a right triangle, an angle cannot be larger than 90° (as the sum of all angles is 180°). So, whenever you’ve been given an angle greater than 90° in a trigonometry problem, draw it in the coordinate plane.
Now, when θ = 30°, sinθ is 0.5.
Now, what value of θ, will sinθ = -0.5?
In the diagram in the previous page, we can see sinθ is negative in the third and fourth quadrant, so there are two possible values of θ i.e.
θ = 180° + 30° & θ = 360° - 30°
θ = 210° & θ = 330°
Solved Example 4
In a right triangle, one angle measures x°, where \(sin\: x^{\circ}-\frac{4}{5}\) . What is cos(90°- x°)
Therefore, cos(90°- x°)= 4/5 or 0.8.
However, you can also solve this problem by constructing a diagram using the given information. It’s a right triangle (which it has to be to use sine/cosine), and the sine of angle x is 4/5.
If \(sin=\frac{Opposite\: Side}{Hypotenuse}\) then the opposite side is 4 long, and the hypotenuse is 5 long:
Since two of the angles of the triangle are of measure x°and 90°, the third angle must have the measure 180° − 90° − x° = 90° − x°
From the figure, cos(90°−x°), which is equal the, is also \(\frac{Adjacent\: Side}{Hypotenuse}\) or 0.8.
For a function, y=sin(Bx), the formula to calculate period is: \(P=\frac{2\pi }{ \left | B \right |}\)
Therefore, the period of y = sin(3x) is \(\frac{2\pi }{3}\)
The above formula is also true for cos.
As done in the functions chapter, the general transformed form of a trigonometric function is:
y = A f(Bx+C) + D
where f is any trigonometric function, A is amplitude, B is used to find the period using equation , C can be used to find the phase shift using equation \(\frac{Original\: Period}{\left | B \right |}\) and D is a vertical shift.
Changing the values of A and D bring vertical changes in the graph while changing the values of B and C will bring the horizontal changes.
The rich text element allows you to create and format headings, paragraphs, blockquotes, images, and video all in one place instead of having to add and format them individually. Just double-click and easily create content.
A rich text element can be used with static or dynamic content. For static content, just drop it into any page and begin editing. For dynamic content, add a rich text field to any collection and then connect a rich text element to that field in the settings panel. Voila!
Headings, paragraphs, blockquotes, figures, images, and figure captions can all be styled after a class is added to the rich text element using the "When inside of" nested selector system.
COORDINATE GEOMETRY is the study of geometry on a plane using a coordinate system. This plane is called a COORDINATE PLANE, which has scales of measurement along the x and y-axes.
SOLID GEOMETRY questions on the SAT Math section are some of the hardest questions you will see on the SAT Math section.
The polygons tested on the SAT are usually the following quadrilaterals: trapezoids, parallelograms, and special parallelograms, such as rhombuses, rectangles, and squares.
Circle questions on the SAT are straightforward and formula driven. The following chapters will list all the formulas you need to know to answer the circle questions on the SAT Math section.
The word TRIGONOMETRY is Greek; tri means three, gon means angle and meter means to measure. Combining all these three words, we get, “three angle measure”. The logic behind this is that trigonometry involves the measurements inside a triangle, which has three angles.
The SAT is absolutely obsessed with triangle-based problems, and as a result, the SAT Math section includes many questions on the SAT Math section. The following set of formulas will help eradicate your trouble once and for all.
There are only a couple of Lines and Angles questions tested on the SAT Math section. The questions are usually easy and should not pose a challenge to you.
.svg)