**What Trigonometry and Calculus Standards and Skills should you master for the Praxis 5161 Exam?**

Since the Praxis 5161 Math Credential enables one teach High School Math, it's an extremely profitable exercise to scrutinize the Math content standards applicable for High School Math teachers in your state.. These are expectations for students that every current and prospective Math teacher ought to be familiar with.

The following content standards apply for Calculus / Trigonometry, in general.

The following content standards apply for Calculus / Trigonometry, in general.

__TRIGONOMETRY__**:**Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric functions studied are defined geometrically rather than in terms of algebraic equations. Facility with these functions as well as the ability to prove basic identities regarding them is especially important for students intending to study calculus, more advanced mathematics, physics and other sciences, and engineering in college.**Candidates shall**- understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians.
- know the definition of sine and cosine as y- and x- coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions.
- know the identity cos2 (x) + sin2 (x) = 1
- prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).
- prove other trigonometric identities and simplify others by using the identity cos2 (x) + sin2 (x) = 1. For example, students use this identity to prove that sec2 (x) = tan2 (x) + 1.
- graph functions of the form f(t) = A sin (Bt + C ) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift.
- know the definitions of the tangent and cotangent functions and can graph them.
- know the definitions of the secant and cosecant functions and can graph them.
- know that the tangent of the angle that a line makes with the x- axis is equal to the slope of the line.
- know the definitions of the inverse trigonometric functions and can graph the functions.
- compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.
- demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/ or simplify other trigonometric identities.
- demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/ or simplify other trigonometric identities.
- use trigonometry to determine unknown sides or angles in right triangles.
- know the law of sines and the law of cosines and apply those laws to solve problems.
- determine the area of a triangle, given one angle and the two adjacent sides.
- are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa.
- represent equations given in rectangular coordinates in terms of polar coordinates.
- are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form.
- know DeMoivre's theorem and can give n th roots of a complex number given in polar form.
- are adept at using trigonometry in a variety of applications and word problems.

__CALCULUS__**:**When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses. These standards outline a complete college curriculum in one variable calculus. Many high school programs may have insufficient time to cover all of the following content in a typical academic year. For example, some districts may treat differential equations lightly and spend substantial time on infinite sequences and series. Others may do the opposite. Consideration of the College Board syllabi for the Calculus AB and Calculus BC sections of the Advanced Placement Examination in Mathematics may be helpful in making curricular decisions. Calculus is a widely applied area of mathematics and involves a beautiful intrinsic theory. Students mastering this content will be exposed to both aspects of the subject.**Candidates shall**- demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity.
- know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity
- prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
- prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.
- demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
- demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.
- demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability
- demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
- demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.
- understand the relation between differentiability and continuity.
- derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
- know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
- find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
- compute derivatives of higher orders.
- know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule.
- use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.
- use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.
- use differentiation to solve related rate problems in a variety of pure and applied contexts.
- know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.
- apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
- demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.
- use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.
- compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.
- know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
- compute, by hand, the integrals of rational functions by combining the techniques with the algebraic techniques of partial fractions and completing the square.
- compute the integrals of trigonometric functions
- understand improper integrals as limits of definite integrals.
- demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.
- know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.

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