Subscribe to our YouTube channel for the latest videos, updates, and tips.


Worksheet on Elimination of Unknown Angle(s) Using Trigonometric Identities

In Worksheet on elimination of unknown angle(s) using Trigonometric identities we will prove various types of practice questions on Trigonometric identities.

Here you will get 11 different types of elimination of unknown angle using Trigonometric identities questions with some selected questions hints.

1. Eliminate θ (theta) in each of the following:

(i) x = a sec θ, y = b tan θ

(ii) a sin θ = p, b tan θ = q

(iii) sin θ + cos θ = m, tan θ + cot θ = n

(iv) sin θ – cos θ = m, sec θ - csc θ = b

2. If sin θ + cos θ = m and sec θ + csc θ = n, then prove that

n(m2 – 1) = 2m.

Hint: n = sec θ + csc θ

    ⟹ n = \(\frac{1}{cos θ}\) + \(\frac{1}{sin θ}\) 

    ⟹ n = \(\frac{sin θ + cos θ}{sin θ cos θ}\) 

    ⟹ n = \(\frac{m}{sin θ cos θ}\) 

    ⟹ sin θ cos θ = \(\frac{m}{n}\) ......... (i) 

Now, m2 – 1 = (sin θ + cos θ)2 - 1 

                   = (sin2 θ + sin2 θ + 2 sin θ cos θ) - 1 

                   = 1 + 2 sin θ cos θ - 1 

                   = 2 sin θ cos θ

                   = 2\(\frac{m}{n}\), From (i)


3. If l1 cos θ + m1 sin θ + n1 = 0 and l2 cos θ + m2 sin θ + n2 = 0 then prove that

(m1n2 – n1m2)2 + (n1l2 – n2l1)2 = (l1m2 – l2m1)2


4. If a sin2 ϕ + b cos2 ϕ = c and p sin2 ϕ + q cos2 ϕ = r then prove that

(b – c)(r – p) = (c – a)(q – r).

Hint: \(\frac{b - c}{c - a}\) = \(\frac{b - (a sin^{2} ϕ + b cos^{2} ϕ)}{(a sin^{2} ϕ + b cos^{2} ϕ) - a}\)

              = \(\frac{(b - a) sin^{2} ϕ}{(b - a) cos^{2} ϕ}\)

                 = tan2 ϕ.

Similarly, \(\frac{q - r}{r - p}\) = \(\frac{q - (p sin^{2} ϕ + q cos^{2} ϕ)}{(p sin^{2} ϕ + q cos^{2} ϕ) - p}\)

                      = \(\frac{(q - p) sin^{2} ϕ}{(q - p) cos^{2} ϕ}\)

                      = tan2 ϕ.

Therefore, \(\frac{b - c}{c - a}\) = \(\frac{q - r}{r - p}\).



5. If a sec θ + b tan θ + c = 0 and a’ sec θ + b’ tan θ + c’ = 0 then prove that

(bc’ – b’c)2 – (ca’ – ac’)2 = (ab’ – a’b)2.


6. If \(\frac{x}{a cos θ}\) = \(\frac{y}{b sin θ}\) and \(\frac{ax}{cos θ}\) - \(\frac{by}{sin θ}\) = a2 – b2, prove that

\(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1.

Hint: \(\frac{x}{cos θ}\) ∙ b - \(\frac{y}{sin θ}\) ∙ a + 0 = 0 and \(\frac{x}{cos θ}\) ∙ a - \(\frac{y}{sin θ}\) ∙ b - (a2 - b2) = 0.

By cross multiplication, \(\frac{\frac{x}{cos θ}}{a(a^{2} - b^{2})}\) = \(\frac{\frac{y}{sin θ}}{b(a^{2} - b^{2})}\) = \(\frac{1}{(a^{2} - b^{2})}\)

⟹ \(\frac{x}{a}\) = cos θ,  \(\frac{y}{b}\) = sin θ. Square these and add.


7. If tan A + sin A = m and tan A - sin A = n then prove that

m2 – n2 = 4 \(\sqrt{mn}\).


8. If x sin3 A + y cos3 A = sin A ∙ cos A and x sin A – y cos A = 0 then prove that

x2 + y2 = 1.

Hint: x sin A - y cos A = 0 

⟹ tan A = \(\frac{y}{x}\)

Again, x ∙ \(\frac{sin^{2} A}{cos A}\) + y ∙ \(\frac{cos^{2} A}{sin A}\) = 1

⟹ x ∙ \(\frac{y}{x}\) sin A + y ∙ \(\frac{x}{y}\) cos A = 1

⟹ x cos A + y sin A = 1

Now, (x sin A - y cos A)2 + (x cos A + y sin A)2 = 02 + 12


9. If csc β – sin β = m3; sec β – cos β = n3 then prove that,

m2n2(m2 + n2) = 1.

Worksheet on Elimination of Unknown Angle(s) Using Trigonometric Identities

10. If a = r cos θ cos β, b = r cos θ sin β and c = r sin θ then prove that,

a2 + b2 + c2 = r2.


11. If p = a sec A cos B, q = b sec A sin B and r = c tan A then prove that, 

\(\frac{p^{2}}{a^{2}}\) + \(\frac{q^{2}}{b^{2}}\) - \(\frac{r^{2}}{c^{2}}\) = 1.



Answers


1. (i) \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1.

(ii) \(\frac{a^{2}}{p^{2}}\) - \(\frac{b^{2}}{q^{2}}\) = 1.

(iii) n(m2 – 1) = 2

(iv) b(1 – a2) = 2a






10th Grade Math

From Worksheet on Elimination of Unknown Angle(s) Using Trigonometric Identities to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Average Word Problems | Worksheet on Average Questions with Answers

    May 20, 25 05:40 PM

    In average word problems we will solve different types of problems on basic concept of average. 1. Richard scored 80, 53, 19, 77, 29 and 96 runs in 6 innings in a series. Find the average runs scored…

    Read More

  2. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  3. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  4. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  5. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More